Mathematical modelling of molecular transmembrane transport and changes of tissues ...lbk.fe.uni-lj.si/wp-content/uploads/2018/09/Disertacija... · 2018-09-10 · PREFACE . The present

University of Ljubljana

Faculty of Electrical Engineering

Janja Dermol-Černe

Mathematical modelling of molecular transmembrane transport

and changes of tissues' dielectric properties due to electroporation

DOCTORAL DISSERTATION

Advisor: Prof. Damijan Miklavčič, Ph.D.

Co-advisor: Prof. Gregor Serša, Ph.D.

Ljubljana, 2018

Univerza v Ljubljani

Fakulteta za elektrotehniko

Janja Dermol-Černe

Matematično modeliranje sprememb električnih lastnosti tkiv in

transporta preko celične membrane pri elektroporaciji

DOKTORSKA DISERTACIJA

Mentor: prof. dr. Damijan Miklavčič

Somentor: prof. dr. Gregor Serša

Ljubljana, 2018

“Do. Or do not. There is no try.”

Master Yoda

PREFACE

The present doctoral dissertation is a result of experimental work and numerical modeling related to

electroporation of single cells, cell suspensions, and tissues. The work was carried out during the Ph.D. study

period at the University of Ljubljana, Faculty of Electrical Engineering, Laboratory of Biocybernetics. The

results of the research work are present in six papers, five of them published and one submitted to international

scientific journals, and in one book chapter.

Paper 1 J. Dermol and D. Miklavčič, “Predicting electroporation of cells in an inhomogeneous

electric field based on mathematical modelling and experimental CHO-cell

permeabilization to propidium iodide determination,” Bioelectrochemistry, vol. 100, pp.

52–61, Dec. 2014.

Paper 2 J. Dermol and D. Miklavčič, “Mathematical Models Describing Chinese Hamster Ovary

Cell Death Due to Electroporation In Vitro,” Journal of Membrane Biology, vol. 248, no.

5, pp. 865–881, Oct. 2015.

Paper 3 J. Dermol-Černe, D. Miklavčič, M. Reberšek, P. Mekuč, S.M. Bardet, R. Burke, D. Arnaud-

Cormos, P. Leveque, and R. O'Connor, “Plasma membrane depolarization and

permeabilization due to electric pulses in cell lines of different excitability,”

Bioelectrochemistry, vol. 122, pp. 103–114, Aug. 2018 .

Paper 4 J. Dermol, O. N. Pakhomova, A. G. Pakhomov, and D. Miklavčič, “Cell

Electrosensitization Exists Only in Certain Electroporation Buffers,” PLOS ONE, vol.

11, no. 7, p. e0159434, Jul. 2016.

Paper 5 J. Dermol-Černe and D. Miklavčič, “From cell to tissue properties – modeling skin

electroporation with pore and local transport region formation,” IEEE Transactions on

Biomedical Engineering, vol. 65, no. 2, pp. 458-468, Feb. 2018.

Paper 6 J. Dermol-Černe, J. Vidmar, J. Ščančar, K. Uršič, G. Serša, and D. Miklavčič, “Connecting

the in vitro and in vivo experiments in electrochemotherapy: Modeling cisplatin

transport in mouse melanoma by the dual-porosity model,” Journal of Controlled

Release, submitted.

Book chapter: J. Dermol and D. Miklavčič, “Mathematical Models Describing Cell Death Due to

Electroporation,” in Handbook of Electroporation, D. Miklavčič, Ed. Cham: Springer

International Publishing, 2016, pp. 1–20.

Acknowledgments

I am deeply grateful to my mentor, Prof. Dr. Damijan Miklavčič for his guidance, his continuous support during

my Ph. D. study, his patience, motivation, and immense knowledge. Thank you for giving me the option to work

in the laboratory already as an undergraduate student, showing me first steps into the scientific world and giving

me the opportunity to work on many interesting projects.

I would also like to thank my Co-Advisor, Prof. Dr. Gregor Serša for help with the in vivo experiments and

interpretation of the data.

I would like to thank my colleagues, ex-colleagues, and collaborators of the Laboratory for Biocybernetics. I am

thankful to Lea and Duša for teaching me how to work with cell cultures and their help with experiments in the

cell culture laboratory. I would like to thank Tadeja, Matej R., Matej K., Bor, Lea R., Samo, Tadej, Barbara,

Gorazd, Alenka, Dan, Vitalij, Olga, and Andrei for stimulating discussions and insightful comments.

I would like to thank Dr. Janja Vidmar, Dr. Janez Ščančar and members of the Jožef Stefan Institute, Department

of environmental physics who measured the cisplatin samples with the inductively coupled plasma – mass

spectrometry.

I would like to thank Katja Uršič and members of the Institute of Oncology Ljubljana, Department of

Experimental Oncology, who performed the in vivo experiments of electrochemotherapy.

I would like to thank Dr. Rodney O’Connor and Dr. Sylvia Bardet Coste for showing a new universe of excitable

cells and ion channels.

I would like to thank Mija Bernik for linguistic proofreading of the abstract in the Slovene language.

And last but not least. I am sincerely grateful to my family for always being there for me and showing me that

knowledge is a precious thing and Matevž for his constant support and for reminding me that I can do it.

This work has been supported by the Slovenian Research Agency (ARRS) under a Junior Research grant and by

research core funding nos. P2-0249 and P3-0003. Experiments were performed within the Infrastructure

Program: Network of research infrastructure centers at the University of Ljubljana (MRIC UL IP-0510). Part of

the work was performed at the Frank Reidy Research Center for Bioelectrics in Norfolk, USA during the Short

Term Scientific Mission STSM-TD 1104-20915 granted by the COST Action TD 1104 (www.electroporation.net)

and in LABEX Sigma-Lim – XLIM, Faculté des Sciences et Techniques ABEX Sigma-Lim – XLIM, Faculté des

Sciences et Techniques, Limoges, France in the scope of the ARRS bilateral project Comparison of

Electroporation Using Classical and Nanosecond Electric Pulses on Excitable and Non-excitable Cells and

Tissues, BI-FR/16-17-PROTEUS. The research was conducted in the scope of the European Associated

Laboratory for Pulsed Electric Field Applications in Biology and Medicine (LEA EBAM).

Table of Contents

Table of Contents Table of Contents .................................................................................................................................................... 1

Abstract ................................................................................................................................................................... 3

Razširjen povzetek v slovenskem jeziku ................................................................................................................. 5

Uvod ................................................................................................................................................................... 5

Elektroporacija ............................................................................................................................................... 5

Uporaba elektroporacije ................................................................................................................................. 5

Medicinski posegi z elektroporacijo – elektrokemoterapija, netermično odstranjevanje tkiva z ireverzibilno

elektroporacijo in vnos učinkovin v kožo in skoznjo ..................................................................................... 6

Načrtovanje posegov elektrokemoterapije in netermičnega odstranjevanja tkiva z ireverzibilno

elektroporacijo ................................................................................................................................................ 7

Električne lastnosti celic in tkiv ..................................................................................................................... 9

Transport majhnih molekul v celice in v tkivo ............................................................................................. 10

Modeliranje elektroporacije.......................................................................................................................... 11

Namen ............................................................................................................................................................... 15

Rezultati in razprava ......................................................................................................................................... 16

Matematični modeli prepustnosti celične membrane, celične smrti in depolarizacije .................................. 16

Prehod modeliranja z ravni ene same celice na raven tkiv z upoštevanjem njegove strukture ..................... 24

Modeliranje transporta molekul skozi celično membrano ............................................................................ 28

Zaključek .......................................................................................................................................................... 31

Izvirni prispevki k znanosti............................................................................................................................... 36

Matematično modeliranje prepustnosti celične membrane in celične smrti ................................................. 36

Prehod modeliranja z nivoja ene same celice na nivo tkiv, pri čemer je predstavljen realističen

tridimenzionalen model kože nadgrajen z modelom elektroporacije vseh bistvenih delov in sestavnih plasti

...................................................................................................................................................................... 36

Matematično modeliranje transporta molekul (kemoterapevtika) skozi celično membrano na osnovi modela

dvojne poroznosti ......................................................................................................................................... 36

Introduction ........................................................................................................................................................... 37

Electroporation – mechanisms, and use ............................................................................................................ 37

Treatment planning of electrochemotherapy and ablation with irreversible electroporation ............................ 37

Dielectric properties of electroporated cells ..................................................................................................... 38

Transport during and after electroporation ....................................................................................................... 39

1

Table of Contents

Models of electroporation ................................................................................................................................. 40

Models at the level of molecules .................................................................................................................. 40

Models at the level of lipid bilayers ............................................................................................................. 40

Models at the level of single cells ................................................................................................................ 41

Models at the level of cell clusters and tissues ............................................................................................. 41

Aim ..................................................................................................................................................................... 44

Results and Discussion ........................................................................................................................................ 45

Paper 1 ............................................................................................................................................................ 49

Paper 2 ............................................................................................................................................................ 61

Book chapter ................................................................................................................................................... 79

Paper 3 ............................................................................................................................................................ 101

Paper 4 ............................................................................................................................................................ 115

Paper 5 ............................................................................................................................................................ 137

Paper 6 ............................................................................................................................................................ 149

Conclusions and future work .............................................................................................................................. 175

Original Scientific Contributions ........................................................................................................................ 180

References ........................................................................................................................................................... 182

2

Abstract

Abstract Electroporation is a phenomenon, which occurs when short high voltage pulses are applied to cells and tissues

resulting in a transient increase in membrane permeability or cell death, presumably due to pore formation. If

cells recover after pulse application, this is reversible electroporation. If cells die, this is irreversible

electroporation. Electroporation is used in biotechnology for biocompound extraction and cryopreservation, in

food processing for sterilization and pasteurization of liquid food and in medicine for treating tumors by

electrochemotherapy or irreversible electroporation as an ablation technique, for gene electrotransfer,

transdermal drug delivery, DNA vaccination, and cell fusion.

In electroporation-based medical treatments, we can treat tumors with predefined electrode geometry and

parameters of electric pulses. When we treat larger tumors of irregular shape treatment plan of the position of the

electrodes and parameters of the electric pulses has to be calculated before each treatment to assure coverage of

the tumor with a sufficient electric field. In treatment plans, currently, 1) we assume that above an

experimentally determined critical electric field all cells are affected and below not, although, in reality, the

transition between non-electroporated and electroporated state is continuous. 2) We do not take into account the

excitability of some tissues. 3) The increase in tissues’ conductivity is described phenomenologically and does

not include mechanisms of electroporation. 4) Transport of chemotherapeutics into the tumor cells in

electrochemotherapy treatments is not included in the treatment plan although it is vital for a successful

treatment. We focused on the mathematical and numerical models of electroporation with the aim of including

them in the treatment planning of electroporation-based medical treatments.

We aimed to model processes happening during electroporation of tissues, relevant in the clinical procedures, by

taking into account processes happening at the single cell level. First, we used mathematical models of cell

membrane permeability and cell death which are phenomenological descriptions of experimental data. The

models were chosen on the basis of the best fit with the experimental data. However, they did not include

mechanisms of electroporation, and their transferability to tissues was questionable. We modeled time dynamics

of dye uptake due to increased cell membrane permeability in several electroporation buffers with regard to the

electrosensitization, i.e., delayed hypersensitivity to electric pulses caused by pretreating cells with electric

pulses. We also modeled the strength-duration depolarization curve and cell membrane permeability curve of

excitable and non-excitable cell lines which could be used to optimize pulse parameters to achieve maximal drug

uptake at minimal tissue excitation.

Second, we modeled change in dielectric properties of tissues during electroporation. Model of change in

dielectric properties of tissues was built for skin and validated with current-voltage measurements. Dielectric

properties of separate layers of skin before electroporation were determined by taking into account geometric

and dielectric properties of single cells, i.e., keratinocytes, corneocytes. Dielectric properties of separate layers

during electroporation were obtained from cell-level models of pore formation on single cells of lower skin

layers (keratinocytes in epidermis and lipid spheres in papillary dermis) and local transport region formation in

the stratum corneum. Current-voltage measurements of long low-voltage pulses were accurately described taking

into account local transport region formation, pore formation in the cells of lower layers and electrode

polarization. Voltage measurements of short high-voltage pulses were also accurately described in a similar way

3

Abstract

as with long low-voltage pulses; however, the model underestimated the current, probably due to

electrochemical reactions taking place at the electrode-electrolyte interface.

Third, we modeled the transport of chemotherapeutics during electrochemotherapy in vivo. In

electrochemotherapy treatments, transport of chemotherapeutics in sufficient amounts into the cell is vital for a

successful treatment. We performed experiments in vitro and measured the intracellular platinum mass as a

function of pulse number and electric field by inductively coupled plasma – mass spectrometry. Using the dual-

porosity model, we calculated the in vitro permeability coefficient as a function of electric field and number of

applied pulses. The in vitro determined permeability coefficient was then used in the numerical model of mouse

melanoma tumor to describe the transport of cisplatin to the tumor cells. We took into account the differences in

the transport of cisplatin in vitro and in vivo caused by the decreased mobility of molecules and decreased

membrane area available for the uptake in vivo due to the high volume fraction of cells, the presence of cell

matrix and close cell connections. Our model accurately described the experimental results obtained in

electrochemotherapy of tumors and could be used to predict the efficiency of electrochemotherapy in vitro thus

reducing the number of needed animal experiments.

In the thesis, we connected the models at the cell level to the models at the tissue level with respect to cell

membrane permeability and depolarization, cell death, change in dielectric properties and transport. Our models

offer a step forward in modeling and understanding electroporation at the tissue level. In future, our models

could be used to improve treatment planning of electroporation-based medical treatments.

4

Razširjen povzetek v slovenskem jeziku - Uvod

Razširjen povzetek v slovenskem jeziku

Uvod

Elektroporacija

Visokonapetostni električni pulzi povečajo prepustnost celične membrane (Tsong 1991; Weaver 1993;

Kotnik et al. 2012) skozi pore (Abidor et al. 1979), ki nastanejo na tistih njenih delih, kjer vsiljena

transmembranska napetost preseže kritično vrednost (Towhidi et al. 2008; Kotnik et al. 2010).

Elektroporacija je reverzibilna, če si celica po pulzih opomore, in ireverzibilna, če je škoda preobsežna in

celica odmre (Pakhomova et al. 2013b; Jiang et al. 2015a). Trenutne optične metode por ne morejo zaznati,

zato njihov nastanek zaznavamo posredno, bodisi z meritvami vnosa različnih molekul v celice ali z

meritvami električnih lastnosti celic (Napotnik in Miklavčič 2017).

Uporaba elektroporacije

V živilski industriji (Toepfl 2012; Toepfl et al. 2014) uporabljamo elektroporacijo oziroma pulzirajoča

električna polja (angl. pulsed electric fields), kar je uveljavljen izraz v tej industriji, za uničevanje patogenih

organizmov in njihovih produktov (encimov in toksinov). V nasprotju s termično obdelavo hrane električni

pulzi ne vplivajo na okus, barvo ali hranilno vrednost. V biotehnologiji uporabljamo elektroporacijo za

ekstrakcijo molekul iz mikroorganizmov in rastlin, s čimer se izognemo uporabi kemičnih sredstev in ne

uničimo celičnih organelov, torej se izognemo tudi dodatnemu čiščenju končnega produkta (Sack et al.

2010; Haberl et al. 2013a; Mahnič-Kalamiza et al. 2014b; Kotnik et al. 2015). Primeri: ekstrakcija DNK iz

bakterij; sladkorja iz sladkorne pese (Haberl et al. 2013b), sokov iz sadja; polifenolov iz grozdja za

izboljšanje kvalitete vina (Puértolas et al. 2010); vode pri sušenju zelene biomase, ki služi kot vir za

biogorivo (Golberg et al. 2016). Elektroporacija je tudi nova metoda pri zamrzovanju celic in tkiv, angl.

cryopreservation (Galindo in Dymek 2016; Dovgan et al. 2017).

Elektroporacijo uporabljamo tudi v medicini (Miklavčič et al. 2010; Yarmush et al. 2014), in sicer pri

elektrokemoterapiji (Miklavčič et al. 2012; Mali et al. 2013; Cadossi et al. 2014; Miklavčič et al. 2014;

Campana et al. 2014; Serša et al. 2015), netermičnem odstranjevanju tkiva z ireverzibilno elektroporacijo

(Davalos et al. 2005; Garcia et al. 2010; José et al. 2012; Cannon et al. 2013; Scheffer et al. 2014b; Jiang et

al. 2015a; Rossmeisl et al. 2015), genski terapiji (Golzio et al. 2002; Vasan et al. 2011; Gothelf in Gehl

2012; Calvet et al. 2014; Heller in Heller 2015; Trimble et al. 2015) in vnosu učinkovin v kožo in skoznjo

(Denet et al. 2004; Zorec et al. 2013b). Pri genski terapiji vnesemo v celice plazmide, v katerih je zapisana

sinteza določenega proteina, ki lahko spremeni biološko funkcijo celice (Aihara in Miyazaki 1998; Heller in

Heller 2015). Z elektroporacijo povišamo varnost genske terapije, saj se izognemo uporabi virusov in

kemikalij. Mehanizmi genske terapije z elektroporacijo še niso popolnoma pojasnjeni, osnovni koraki so

opisani v literaturi (Rosazza et al. 2016). Z elektroporacijo lahko zlivamo različne celice, s čimer

pridobivamo celice, ki proizvajajo monoklonska protitelesa ali inzulin (Ramos in Teissié 2000; Trontelj et

al. 2008; Rems et al. 2013).

5


V doktorski disertaciji sem se osredotočila na uporabo elektroporacije v medicini, predvsem pri

elektrokemoterapiji, netermičnem odstranjevanju tkiva z ireverzibilno elektroporacijo in pri vnosu

učinkovin v kožo in skoznjo je, zato so ti trije posegi podrobneje opisani v naslednjem poglavju.

Medicinski posegi z elektroporacijo – elektrokemoterapija, netermično odstranjevanje tkiva z

ireverzibilno elektroporacijo in vnos učinkovin v kožo in skoznjo

Elektrokemoterapija je kombinacija kemoterapije in električnih pulzov, dovedenih neposredno na tarčno

tkivo. Električni pulzi povečajo prepustnost celične membrane za kemoterapevtike, zato povečamo

učinkovitost zdravljenja, obenem pa zmanjšamo dovedeno dozo kemoterapevtika in omilimo stranske

učinke. Celoten tumor mora biti pokrit z dovolj visokim električnim poljem, da povečamo prepustnost vseh

tumorskih celic (Miklavčič et al. 2006a), zagotoviti pa moramo tudi dovolj visoko koncentracijo

kemoterapevtika znotraj tumorja (Miklavčič et al. 2014). Okoliško tkivo ne sme biti uničeno, torej mora biti

električno polje okoli tumorja pod mejo za ireverzibilno elektroporacijo. Pri elektrokemoterapiji običajno

dovajamo osem pulzov dolžine 100 μs s ponavljalno frekvenco 1 Hz. S poskusi določena meja za povišanje

prepustnosti tumorskega tkiva je 0,4 kV/cm (Miklavčič et al. 2010). Osem pulzov je bilo določenih kot

optimalno število pulzov (Marty et al. 2006; Mir et al. 2006), večje število dovedenih pulzov namreč že

zmanjšuje preživetje (Dermol in Miklavčič 2015). Za zdravljenje tumorjev z elektrokemoterapijo so bili

definirani standardni postopki (angl. standard operating procedures) (Marty et al. 2006; Mir et al. 2006),

kjer so glede na število tumorjev, njihovo velikost in lokacijo (na koži ali pod kožo) določeni tip elektrod,

kemoterapevtik, anestezija in način dovajanja kemoterapevtika. Kemoterapevtik lahko dovedemo lokalno

ali sistemsko. V elektrokemoterapiji oz. terapiji z električnimi pulzi sta najbolj razširjena kemoterapevtika

cisplatin in bleomicin. Z elektrokemoterapijo je možno zdraviti tudi globlje ležeče tumorje (Miklavčič et al.

2010; Pavliha et al. 2013; Edhemović et al. 2014; Miklavčič in Davalos 2015). V zadnjem času se

uveljavlja tudi uničevanje tumorskih celic z visokimi koncentracijami kalcija in električnimi pulzi

(Frandsen et al. 2015; Frandsen et al. 2016; Frandsen et al. 2017). Pri elektrokemoterapiji se pojavijo še

dodatni učinki, ki povišajo učinkovitost elektroporacije. Vazokonstrikcija zmanjša spiranje kemoterapevtika

iz tumorja in s tem ohranja visoko koncentracijo kemoterapevtika v tumorju, obenem se zmanjša pretok

krvi skozi tumor, kar povzroči hipoksijo in pomanjkanje hranilnih snovi (Mir 2006; Serša et al. 2008).

Elektrokemoterapija sproži tudi odziv imunskega sistema, ki nato odstrani preostale tumorske celice (Serša

et al. 2015).

Z ireverzibilno elektroporacijo netermično odstranjujemo tumorje brez uporabe kemoterapevtika (Jiang et

al. 2015a). Tako se popolnoma izognemo stranskim učinkom kemoterapevtikov, vendar na račun več

dovedene energije in posledično Joulovega gretja. Pri ireverzibilni elektroporaciji dovajamo več (okoli 90)

električnih pulzov, dolgih od 50 μs do 100 μs, s ponavljalno frekvenco 1 Hz. Dovedeno električno polje je v

rangu nekaj kV/cm, kar je dosti več kot pri elektrokemoterapiji. Pri ireverzibilni elektroporaciji lahko z

visoko natančnostjo odstranimo želeno tkivo – območje med uničenim in nepoškodovanim tkivom je široko

le nekaj premerov celic (Rubinsky et al. 2007). Za odstranjevanje tumorjev tradicionalno uporabljamo

termične metode (Hall et al. 2014) – radiofrekvenčno odstranjevanje in odstranjevanje s tekočim dušikom,

kjer tkivo uničujemo z visoko oz. z nizko temperaturo. Prednost ireverzibilne elektroporacije pred

uveljavljenimi termičnimi metodami je krajši čas zdravljenja, izognemo se učinkom hlajenja oz. gretja tkiva

6


zaradi bližine žil (Golberg et al. 2015), pri čemer ostanejo okoliške pomembne strukture (žile, živci)

nedotaknjene (Jiang et al. 2015a). Tudi pri ireverzibilni elektroporaciji je v dokončno odstranitev tumorskih

celic vpleten imunski sistem (Neal et al. 2013).

Pri elektrokemoterapiji in ireverzibilni elektroporaciji se zaradi daljših pulzov in ponavljalne frekvence

1 Hz pojavljajo težave zaradi krčenja mišic (Miklavčič et al. 2005), bolečine med dovajanjem pulzov,

heterogenosti električnih lastnosti tkiv v tem frekvenčnem področju ter zaradi možnosti srčnih aritmij (Ball

et al. 2010). Bolečini in krčenju mišic se lahko izognemo, če pulze dovajamo z višjo frekvenco, npr. 5 kHz

(Županič et al. 2007; Serša et al. 2010). Srčnim aritmijam se izognemo tako, da s sinhroniziramo dovedene

električne pulze z električno aktivnostjo srčne mišice (Mali et al. 2008; Deodhar et al. 2011a; Mali et al.

2015). Bolečini, krčenju mišic in heterogenosti električnih lastnosti tkiv se lahko izognemo z dovajanjem

1 μs bipolarnih pulzov (Arena et al. 2011; Arena in Davalos 2012; Sano et al. 2015). V zadnjem času so se

pojavile tudi metode, s katerimi so vnos barvil v celico dosegli brezkontaktno s t. i. magnetoporacijo (Chen

et al. 2010; Towhidi et al. 2012; Kardos in Rabussay 2012; Novickij et al. 2015; Kranjc et al. 2016;

Novickij et al. 2017b; Novickij et al. 2017a).

Elektroporacijo lahko uporabljamo ne le za zdravljenje tumorjev, temveč tudi za vnos učinkovin v kožo in

skoznjo. Vnos učinkovin skozi kožo je neinvaziven, poleg tega pa se izognemo degradaciji učinkovin pri

prehodu skozi prebavni trakt. Skozi kožo lahko preide le malo molekul, zato uporabljamo različne metode

za povečanje prehoda učinkovin – iontoforezo, radiofrekvenčno mikroablacijo, laser, mikroigle, ultrazvok

in elektroporacijo (Zorec et al. 2013b). Proces elektroporacije kože je slabo razumljen. Predpostavljamo, da

pri dovajanju visokonapetostnih električnih pulzov v roženi plasti nastanejo lokalna transportna območja,

kjer sta povišani električna prevodnost in prepustnost (Pliquett et al. 1996; Pliquett et al. 1998; Pliquett et

al. 1998; Pavšelj in Miklavčič 2008a). Skozi lokalna transportna območja lahko nato učinkovine še nekaj ur

po dovedenih pulzih vstopajo skozi kožo v krvni obtok (Zorec et al. 2013a). Gostota teh območij je odvisna

od električnega polja v koži – višje električno polje jih povzroči več. Velikost lokalnih transportnih območij

je odvisna od trajanja pulza. Med samim pulzom se zaradi Joulovega gretja topijo lipidi v roženi plasti, kar

povzroči njihovo širjenje (Pliquett et al. 1996; Prausnitz et al. 1996; Pliquett et al. 1998; Weaver et al. 1999;

Vanbever et al. 1999; Gowrishankar et al. 1999b).

Načrtovanje posegov elektrokemoterapije in netermičnega odstranjevanja tkiva z ireverzibilno

elektroporacijo

Pri zdravljenju tumorjev z elektroporacijo lahko uporabimo standardne oblike in postavitve elektrod z že

določenimi parametri električnih pulzov (Marty et al. 2006; Mir et al. 2006; Campana et al. 2014). Če

zdravimo velike tumorje ali tumorje nepravilnih oblik, ki pogosto ležijo globlje, s standardno postavitvijo

elektrod ne moremo zagotoviti ustrezne pokritosti tumorja z dovolj visokim električnim poljem. V tem

primeru lahko elektrode med samim posegom večkrat premaknemo ali pa prilagodimo njihovo število in

postavitev. Pri tem moramo prej pripraviti načrt posega (Kos et al. 2010; Miklavčič et al. 2010; Pavliha et

al. 2012; Linnert et al. 2012; Edhemović et al. 2014). V njem zagotovimo, da bo cel tumor izpostavljen

dovolj visokemu električnemu polju (Miklavčič et al. 2006a), obenem pa škoda na okoliškem tkivu

minimalna.

7


Načrtovanje posega poteka v več korakih: 1. zajem medicinskih slik (računalniška tomografija, magnetna

resonanca) tumorja in okoliškega tkiva; 2. obdelava slik; 3. razgradnja slik in določitev geometrije tkiva; 4.

vzpostavitev tridimenzionalnega modela; 5. optimizacija postavitve elektrod glede na obliko in velikost

tumorja; 6. izdelava modela elektroporacije (izračun električnega polja in spremembe električne

prevodnosti tkiva); 7. optimizacija napetosti med elektrodami in položaja elektrod (Pavliha et al. 2012). Na

sliki 1 lahko vidimo izračunano električno polje v tumorju in okoliškem tkivu pri eni izmed možnih

postavitev elektrod.

Med načrtovanjem posega privzamemo naslednje predpostavke.

1. Pri določanju elektroporiranega in neelektroporiranega območja uporabljamo s poskusi določeno

vrednost kritičnega električnega polja (Miklavčič et al. 2000; Šel et al. 2005; Čorović et al. 2012). Za vsako

tkivo privzamemo dve kritični vrednosti električnega polja – eno za reverzibilno in drugo za ireverzibilno

elektroporacijo. Predpostavljamo, da so celične membrane vseh celic, ki so izpostavljene električnemu

polju pod kritično vrednostjo za reverzibilno elektroporacijo, neprepustne. Vse celice, ki so izpostavljene

električnemu polju nad kritično vrednostjo za reverzibilno elektroporacijo, imajo prepustne membrane.

Tiste, ki so izpostavljene električnemu polju nad kritično vrednostjo za ireverzibilno elektroporacijo, pa so

mrtve. Vendar v resnici kritično električne polje za elektroporacijo ni enako za vse celice v tkivu, temveč je

statistično porazdeljeno. Če pri načrtovanju posegov predpostavljamo, da imajo vse celice enak prag za

reverzibilno in ireverzibilno elektroporacijo, je lahko tumor izpostavljen prenizkemu električnemu polju in

tumorske celice preživijo. Če tkivo izpostavimo previsokemu električnemu polju, lahko poškodujemo

pomembne bližnje strukture.

2. Z elektroporacijo lahko vplivamo tudi na vzdražna tkiva, kot so nevroni in mišice. Zdravimo jih pri

netermičnem odstranjevanju tumorjev v možganih z ireverzibilno elektroporacijo (Garcia et al. 2012;

Rossmeisl et al. 2015; Sharabi et al. 2016), pri genski terapiji (Hargrave et al. 2013; Hargrave et al. 2014;

Bulysheva et al. 2016) ali netermičnem odstranjevanju tkiva srčne mišice z ireverzibilno elektroporacijo

(Neven et al. 2014b; Neven et al. 2014a) in pri genski terapiji skeletnih mišic (Aihara in Miyazaki 1998).

Lahko pa vzdražna tkiva električnemu polju izpostavimo nenamerno, ko so v bližini tarčnega območja.

Vplivamo na živčno-žilni snop pri zdravljenju raka na prostati (Neal et al. 2014; Ting et al. 2016), na

nevrone pri zdravljenju kostnih metastaz (Tschon et al. 2015; Gasbarrini et al. 2015) ali na hrbtenjačo pri

zdravljenju tumorjev v hrbtenici (Tschon et al. 2015). Električna stimulacija živcev in mišic povzroči

nelagodje in bolečino (Miklavčič et al. 2005; Županič et al. 2007; Arena in Davalos 2012; Golberg in

Rubinsky 2012). Pokazali so, da lahko električni pulzi povzročijo stimulacijo receptorjev za bolečino

(nociceptorjev) (Nene et al. 2006; Jiang in Cooper 2011). Pomemben napredek pri zdravljenju vzdražnih

tkiv ali tkiv v njihovi bližini bi bila določitev optimalnih parametrov električnega polja, kjer dosežemo

najvišjo prepustnost celične membrane, obenem pa ne vzdražimo tkiva.

3. Elektroporacija poviša električno prevodnost tkiva zaradi povišane električne prevodnosti celičnih

membran in zaradi Joulovega gretja (Šel et al. 2005; Ivorra et al. 2009; Essone Mezeme et al. 2012a; Neal

et al. 2012). Spremenjena električna prevodnost vpliva na porazdelitev električnega polja (Šel et al. 2005).

Pri načrtovanju posegov predpostavljamo, da se prevodnost tkiva poviša za približno štirikrat (Neal et al.

2012). Tkiva imajo različne električne lastnosti, ki se spremenijo in vplivajo na porazdelitev električnega

8


polja. Vrednosti električnih lastnosti tkiv po elektroporaciji lahko izračunamo z ekvivalentnim vezjem

(Neal et al. 2012), lahko pa bi v izračune vključili tudi mehanizme elektroporacije.

4. Pri elektrokemoterapiji je za učinkovitost zdravljenja najpomembnejši zadosten vnos

kemoterapevtika v celice (Miklavčič et al. 2014). Pri nizki znotrajcelični koncentraciji kemoterapevtika

bleomicina pride do mitotske smrti, pri visoki pa do procesa, podobnega apoptozi (programirani celični

smrti) (Tounekti et al. 1993; Tounekti et al. 2001). V načrtovanju zdravljenja predpostavljamo, da je

električno polje nad kritično vrednostjo za reverzibilno elektroporacijo tkiva dovolj za zadosten vnos

kemoterapevtika v celice. Lahko se zgodi, da kemoterapevtika kljub dovolj visokemu električnemu polju v

celice ne pride dovolj, če je njegova začetna prostorska porazdelitev v tumorju nehomogena in ga v

zunajceličnem prostoru zmanjka.

Slika 1: Primer izračunanega električnega polja v različnih tkivih (tumorju, mišicah, maščobnem tkivu) pri

zdravljenju tumorja z elektroporacijo. Tumor je označen z modro barvo in je cel izpostavljen električnemu

polju nad kritično vrednostjo za reverzibilno elektroporacijo. Povzeto po (Županič et al. 2012).

Električne lastnosti celic in tkiv

Elektroporacija poveča električno prevodnost celične membrane, kar so merili v več raziskavah z

dielektrično impedančno spektroskopijo (Abidor et al. 1993; Schmeer et al. 2004), s tokovnonapetostnimi

meritvami (Pliquett in Wunderlich 1983; Pavlin et al. 2005; Pavlin in Miklavčič 2008; Suzuki et al. 2011), z

optičnimi metodami (Hibino et al. 1991; Griese et al. 2002) ali z metodo vpete krpice, angl. patch clamp

(Pakhomov et al. 2007; Wegner 2015; Yoon et al. 2016; Napotnik in Miklavčič 2017). Zaradi

elektroporacije se poveča tudi električna prevodnost sloja celic (Ghosh et al. 1993; Müller et al. 2003;

Stolwijk et al. 2011; García-Sánchez et al. 2015). Elektroporacija spremeni električne lastnosti človeških,

živalskih in rastlinskih tkiv, kar lahko merimo s podobnimi metodami kot pri celicah – z dielektrično

impedančno spektroskopijo (Barnes in Greenebaum 2006; Ivorra in Rubinsky 2007; Grimnes in Martinsen

2008; Dean et al. 2008; Zhuang et al. 2012; Dymek et al. 2014; Trainito et al. 2015; Zhuang in Kolb 2015),

s tokovnonapetostnimi meritvami (Pavlin et al. 2005; Cukjati et al. 2007; Ivorra in Rubinsky 2007; Pavlin

in Miklavčič 2008; Ivorra et al. 2009; Chalermchat et al. 2010; Neal et al. 2012; Becker et al. 2014; Dymek

et al. 2015), z električno impedančno tomografijo (Davalos et al. 2002; Davalos et al. 2004; Granot in

Rubinsky 2007; Meir in Rubinsky 2014) ali z magnetno resonančno električno impedančno tomografijo

(Kranjc et al. 2014; Kranjc et al. 2015; Kranjc et al. 2017). Elektroporacija spremeni specifično prevodnost

9


tkiva predvsem v območju β-disperzije (Pliquett et al. 1995; Ivorra in Rubinsky 2007; Garner et al. 2007;

Oblak et al. 2007; Ivorra et al. 2009; Neal et al. 2012; Zhuang et al. 2012; Salimi et al. 2013). Ta nastane

zaradi polarizacije celične membrane, na katero elektroporacija najbolj vpliva zaradi por v njej in

posledično povečane prevodnosti. Iz spremembe elektičnih lastnosti tkiv lahko sklepamo na odziv tkiva na

zdravljenje (Cukjati et al. 2007; Ivorra et al. 2009; Neal et al. 2012). Pri meritvah električnih lastnosti po

elektroporaciji moramo biti pozorni na sekundarne učinke elektroporacije, ki lahko vplivajo na rezultate

meritev: spremembo velikosti celic (Serša et al. 2008; Sano et al. 2010; Deodhar et al. 2011b; Calmels et al.

2012), nastanek edema (Ivorra et al. 2009), izpust ionov iz celic (Pavlin et al. 2005) in vaskularno okluzijo

(Ivorra in Rubinsky 2007; Serša et al. 2008).

Specifična prevodnost tkiv s frekvenco narašča, dielektričnost pa pada. Obe lahko razdelimo na več

značilnih disperzijskih območij (Pethig 1984; Pethig in Kell 1987; Gabriel et al. 1996a; Miklavčič et al.

2006b; Barnes in Greenebaum 2006; Grimnes in Martinsen 2008), kar lahko vidimo na sliki 2. Pri nizkih

frekvencah (pod 100 kHz) je prisotna α disperzija zaradi interakcij v bližini celične membrane,

znotrajceličnih struktur, ionske difuzije in polarizacije elektrod. β disperzija je prisotna v območju

megahertzov in nastane zaradi polarizacije celičnih membran in odzivov proteinov. γ disperzija nastane nad

1 GHz zaradi dipolarnih mehanizmov v polarnih medijih, kot je voda. Obstaja tudi manjša δ disperzija pri

nekaj 100 MHz, ki nastane zaradi sprostitve vezane vode in manjših polarnih delov bioloških molekul ter

zaradi ionske difuzije ob nabiti površini (Barnes in Greenebaum 2006). Električne lastnosti tkiv so

prikazane na sliki 2, vendar so absolutne vrednosti različne od tkiva do tkiva (Gabriel et al. 1996a; Gabriel

et al. 1996b; Gabriel et al. 1996c; Grimnes in Martinsen 2008). Obenem so lahko tkiva tudi anizotropna, kar

pomeni, da je tkivo različno električno prevodno v različne smeri. Tipičen primer anizotropnega tkiva so

mišice (Čorović et al. 2010).

Slika 2: Frekvenčna odvisnost električne prevodnosti (σ) in relativne dielektričnosti (εr). Vidimo lahko, da

dielektričnost s frekvenco pada, specifična prevodnost pa narašča. Povzeto po (Miklavčič et al. 2006b).

Transport majhnih molekul v celice in v tkivo

Transport molekul skozi celično membrano zaradi elektroporacije je posledica treh različnih mehanizmov –

difuzije, elektroforeze in endocitoze. Med samimi pulzi je transport elektroforetski in difuzijski (Li in Lin

2011; Sadik et al. 2013), po pulzih pa predvsem difuzijski (Pucihar et al. 2008). Endocitozo so opazili po

dovajanju daljših pulzov pri nižjih električnih poljih (Rols et al. 1995; Antov et al. 2005).

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In vitro lahko transport določamo z različnimi tehnikami (Napotnik in Miklavčič 2017), kot so

fluorescentna mikroskopija, pretočna citometrija, vnos citotoksičnih molekul (Maček Lebar in Miklavčič

2001), vnos plazmidov (Golzio et al. 2002), vnos magnetnih nanodelcev, masna spektrometrija (Čemažar et

al. 2001) ali barvanje z barvili. Pri tehnikah in vivo je merjenje transporta zahtevnejše, saj je treba tkivo

pred analizo večinoma izrezati in razgraditi. Uporabljamo metode, kot so radioaktivno označevanje z

molekulami 57Co-bleomicin (Belehradek et al. 1994), 111In-bleomicin (Engström et al. 1998), 99mTc-DTPA

(Grafström et al. 2006), 51Cr-EDTA (Batiuškaitė et al. 2003), magnetna resonanca z gadolinijem (Garcia et

al. 2012; Kranjc et al. 2015), masna spektrometrija (Melvik et al. 1986; Čemažar et al. 1998; Čemažar et al.

1999; Ogihara in Yamaguchi 2000; Čemažar et al. 2001; Serša et al. 2010; Hudej et al. 2014; Čemažar et al.

2015) in barvanje z barvilom lucifer rumeno (Kranjc et al. 2015).

Pri elektrokemoterapiji je transport ključen za učinkovitost zdravljenja (Miklavčič et al. 2014). Primerjava

transporta in vitro ter in vivo je težavna. Celice v celični suspenziji so obdane s topljencem, ki lahko vanje

vstopi z vseh strani. Transport poteka večinoma z difuzijo. Če je topljenca dovolj, se transport ustavi, ko je

membrana zaceljena. Difuzija po zunajcelični tekočini ni omejena in transport topljenca skozi celično

membrano lahko neposredno povežemo z njeno prepustnostjo (Mahnič-Kalamiza et al. 2014a). In vivo je

začetna koncentracija topljenca prostorsko nehomogena, celice imajo na voljo različno začetno

koncentracijo topljenca, ki ga lahko zmanjka, še preden je membrana zaceljena. Transport topljenca z

difuzijo in konvekcijo (Canatella et al. 2004; Pucihar et al. 2008) poteka z določeno časovno konstanto.

Električni pulzi povzročijo vazokonstrikcijo (Serša et al. 2008), ki omeji izpiranje in pritok topljenca. Tesen

stik med celicami zmanjšuje površino membrane, skozi katero lahko poteka transport. Zunajcelični matriks

ovira transport topljenca. Že pri elektroporaciji sferoidov (preprost približek tkiva) v fluorescentnem barvilu

so ugotovili, da je transport majhnih molekul po sferoidu nehomogen – najvišji vnos so dosegli na robu

sferoida, nato pa se je zmanjševal proti njegovemu središču (Canatella et al. 2004; Gibot et al. 2013).

Vazokonstrikcija po električnih pulzih zmanjša pretok krvi v tumor in iz njega, kar podaljša izpostavljenost

kemoterapevtiku. V tumorju je transport še dodatno otežen zaradi povišanega intersticijskega tlaka,

heterogene prekrvavitve, vezave aktivnih molekul na netarčne molekule v tumorju, okvarjenega limfnega

sistema in metabolizma (Baxter in Jain 1989; Baxter in Jain 1990; Baxter in Jain 1991; Jain 1999).

Modeliranje elektroporacije

Z uporabo matematičnih in fizikalnih zakonov pri modeliranju strnjeno opišemo biološke pojave ter

omogočimo razumevanje in napovedovanje odziva ter zmanjšamo število poskusov. Elektroporacijo

modeliramo na različnih prostorskih ravneh: molekule lipidov, lipidni dvosloj, celice in tkiva. Trenutno so

modeli večinoma omejeni na eno prostorsko raven, povezovalnih modelov med njimi pa je malo.

Modeli na ravni molekul

Osnovni gradniki celične membrane so fosfolipidi. Obnašanje lipidov in drugih molekul pod vplivom

električnega polja lahko ugotavljamo s simulacijami molekularne dinamike (Tieleman 2006; Böckmann et

al. 2008; Delemotte in Tarek 2012; Reigada 2014; Casciola et al. 2014; Kirsch in Böckmann 2015), ki so

zaradi računske zahtevnosti časovno omejene nekaj 100 do 1000 nanosekund. Raziskave molekularne

dinamike predvidevajo, da zaradi električnega polja v lipidnem dvosloju nastanejo hidrofobne pore, ki se

11


razvijejo v hidrofilne pore. Simulacije numerične dinamike so primerjali s poskusi. Ugotovili so, da lahko

mala interferenčna RNA (siRNA) v nekaj nanosekundah prečka celično membrano (Breton et al. 2012).

Modeli na ravni lipidnih dvoslojev

V polarnem topilu, kot je voda, se lipidi spontano uredijo v lipidni dvosloj. Modele elektroporacije lipidnih

dvoslojev lahko razdelimo v dve skupini (Chen et al. 2006a; Pavlin et al. 2008; Rems in Miklavčič 2014). V

prvi so deterministični elektromehanski (hidrodinamični, elastični, hidroelastični, viskoelastični) modeli,

kjer dvosloj obravnavamo kot elastično telo in ga opisujemo z zakoni elektrostatike. Druga skupina so

stohastični modeli nastanka por. Temeljijo na energijskih enačbah (Abidor et al. 1979), združuje pa jih

enačba Smoluchowskega, ki opiše nastanek in dinamiko razvoja por. Neu in Krassowska sta jo poenostavila

in omogočila njen relativno preprost numerični izračun (Neu in Krassowska 1999; Neu in Neu 2009),

vendar pa v njej predvidevata, da se velikost por med pulzom ne spreminja. Asimptotski model

elektroporacije je torej primeren le za kratke nanosekundne pulze, pri daljših pulzih namreč že pride do

širjenja por v celični membrani. Tudi širjenje por med elektroporacijo lahko modeliramo, a je računsko bolj

zahtevno, ker za vsako poro posebej rešujemo diferencialno enačbo spremembe premera v vsaki časovni

točki (Smith et al. 2004). Med energijske modele spada tudi model, ki nastanek por opiše s kinetično shemo

prehajanja membrane (Neumann et al. 1998; Miklavčič in Towhidi 2010) med prepustnimi in

neprepustnimi stanji (Böckmann et al. 2008). Transport majhnih molekul skozi poro lahko opišemo z

elektrodifuzijsko enačbo (Granot in Rubinsky 2008; Li in Lin 2011; Smith in Weaver 2011; Movahed in Li

2012; Movahed in Li 2013). Pri primerjavi zveznih modelov elektroporacije lipidnih dvoslojev in modelov

molekularne dinamike so ugotovili, da se rezultati obojih dobro ujemajo (Casciola et al. 2016; Rems et al.

2016).

Modeli na ravni posameznih celic

Lipidni dvosloj je skupaj z membranskimi proteini osnova za nastanek biološke celice. Biološka celica je v

približku električno slabo prevodna membrana (dielektrik), ki je na obeh straneh obdana z elektrolitom

(prevodnik). Na ravni celic obstajajo modeli različnih pojavov: vsiljene transmembranske napetosti,

nastanka por, transporta, prepustnosti celične membrane, smrti, spremembe prevodnosti. Časovna in

prostorska dinamika vsiljene transmembranske napetosti sta analitično rešljivi za preproste oblike

osamljenih celic (Kotnik et al. 1998; Kotnik in Miklavčič 2000a; Kotnik in Miklavčič 2000b; Valič et al.

2003; Kotnik in Miklavčič 2006), v splošnem pa zahtevata numerični izračun (Pucihar et al. 2006). Ko je

vsiljena transmembranska napetost znana, lahko opišemo nastanek por (Neu in Krassowska 1999; Smith et

al. 2004; Neu in Neu 2009). Celice lahko modeliramo tudi z ekvivalentnim električnim vezjem (Pethig

1984; Gowrishankar in Weaver 2003). Smrt in prepustnost celic v suspenziji lahko opišemo s statističnimi

modeli (Peleg 1995; Huang et al. 2012; Dermol 2014; Dermol in Miklavčič 2014; Dermol in Miklavčič

2015; Dermol in Miklavčič 2016). V literaturi se pojavljajo štirje modeli prepustnosti celične membrane –

simetrična in asimetrična sigmoida, Gompertzova krivulja in hiperbolični tangens (Laird 1964; Puc et al.

2003; Essone Mezeme et al. 2012b; Dermol in Miklavčič 2014). Modelov celične smrti je več (Hülsheger in

Niemann 1980; Hülsheger et al. 1981; Geeraerd et al. 2000; Peleg in Penchina 2000; Huang et al. 2012),

večinoma izhajajo iz živilske industrije, kjer z njimi napovedujejo učinkovitost uničevanja patogenih

organizmov v živilih z namenom pasterizacije in sterilizacije (Peleg 2006; Huang et al. 2012; Dermol in

12


Miklavčič 2016). Električno prevodnost in prepustnost celične membrane lahko modeliramo ločeno

(Miklavčič in Towhidi 2010; Leguèbe et al. 2014). Transport molekul v posamezno celico lahko opišemo z

elektrodifuzijsko enačbo skozi prepustno membrano (Zaharoff et al. 2008; Pucihar et al. 2008; Leguèbe et

al. 2014; Blumrosen et al. 2016).

Modeli na ravni tkiv in skupkov celic

Skupek po obliki in funkciji podobnih celic tvori tkivo (Alberts et al. 2009). Modele tkiv delimo v dve

skupini – tkivo jemljejo kot homogeno enoto (Šel et al. 2005; Čorović et al. 2007; Pavšelj et al. 2007;

Pavšelj in Miklavčič 2008b; Golberg in Rubinsky 2010; Neal et al. 2012; Čorović et al. 2012; Garcia et al.

2014) ali pa vključujejo tudi mikrostrukturo tkiva (Gowrishankar in Weaver 2003; Stewart Jr. et al. 2005;

Gowrishankar in Weaver 2006; Esser et al. 2007; Joshi et al. 2008; Esser et al. 2009; Essone Mezeme et al.

2012b; Dymek et al. 2015).

Modeli homogenega tkiva obravnavajo električno polje v odvisnosti od oblike in števila elektrod,

parametrov električnih pulzov ter lastnosti tkiva in sprememb njegove prevodnosti (Šel et al. 2005; Čorović

et al. 2007; Pavšelj et al. 2007; Pavšelj in Miklavčič 2008b). Za vsak tip tkiva privzamemo s poskusi

določeno kritično vrednost električnega polja, kjer pride do povišane prepustnosti membrane ali do smrti

(Šel et al. 2005; Čorović et al. 2012), kar uporabimo za optimizacijo parametrov električnega polja. Z

matematičnimi modeli lahko napovemo verjetnost za povišano prepustnosti celične membrane ali celično

smrt (Golberg in Rubinsky 2010; Garcia et al. 2014; Garcia et al. 2014; Sharabi et al. 2016; Kranjc et al.

2017). S fenomenološkimi modeli, ki temeljijo na različnih porazdelitvah relaksacijskih časov molekul v

tkivu, opišemo frekvenčno odvisnost električnih lastnosti različnih tkiv (Debyev, Cole-Colejev, Cole-

Davidsonov, Havriliak-Negamijev, Raicujev model (Barnes in Greenebaum 2006)).

Pri elektroporaciji kože modeli opisujejo električno polje ali transport v koži kot organu iz posameznih

homogenih slojev (Chizmadzhev et al. 1998; Weaver et al. 1999; Becker in Kuznetsov 2007; Pavšelj et al.

2007; Pavšelj in Miklavčič 2008a; Becker 2012; Becker et al. 2014). Električne lastnosti slojev pred

elektroporacijo in po njej so večinoma približki, obenem lahko več slojem pripišemo enake lastnosti

(Pavšelj et al. 2007). Nekaj modelov vključuje tudi nastanek lokalnih transportnih območij (Becker in

Kuznetsov 2007; Pavšelj in Miklavčič 2008a; Becker 2012), ampak so omejeni le na procese v roženi plasti

in ne zajemajo elektroporacije celic v spodnjih slojih kože.

Transport snovi po tkivu (Becker in Kuznetsov 2013) lahko dobro opišemo s statističnimi modeli, ki pa ne

vključujejo mehanizmov transporta (Canatella in Prausnitz 2001), s kinetično shemo elektroporacije

(Neumann et al. 1998; Miklavčič in Towhidi 2010), s farmakološkim modelom (Puc et al. 2003; Agarwal et

al. 2009), z elektrodifuzijsko enačbo ali z modelom dvojne poroznosti (Mahnič-Kalamiza et al. 2014a;

Mahnič-Kalamiza et al. 2015). Model dvojne poroznosti temelji na difuzijski enačbi, povezuje nastanek por

s prepustnostjo celične membrane in omogoča modeliranje transporta med več razdelki, npr. celicami,

intersticijsko tekočino in sosednjimi tkivi. V ta model lahko vključimo termične procese (Mahnič-Kalamiza

et al. 2017) ali pa ga povežemo z enačbami za stopnjo elektroporacije, spremembo električne prevodnosti

tkiv in celjenje por (Boyd in Becker 2015; Argus et al. 2017) ter celostno opišemo proces elektroporacije in

transporta.

13


Druga večja skupina modelov tkiv upošteva mikrostrukturo tkiv. Iz električnih lastnosti posameznih

okroglih celic v nizkih prostorninskih deležih lahko izračunamo ekvivalentne lastnosti z enačbami mešanja,

kjer upoštevamo električne lastnosti in prostorninske deleže celic (Asami 2002). Uporabimo enačbe

Maxwell-Wagnerja, Hanaija, Böttcherja, Bruggemana, Looyenge (Pavlin in Miklavčič 2009). V dveh

dimenzijah lahko presek tkiva vzorčimo s kartezijskimi transportnimi mrežami in s Kirchoffovimi zakoni

izračunamo tok, napetost in električni naboj po celi mreži (Gowrishankar in Weaver 2003; Stewart ml. et al.

2005; Gowrishankar in Weaver 2006; Esser et al. 2007; Esser et al. 2009). Podobno metodo ekvivalentnega

vezja so uporabili v literaturi (Ramos et al. 2003; Ramos 2005), kjer so najprej analizirali učinke

elektroporacije na ravni celic, nato pa preko ekvivalentnih električnih lastnosti celic prešli na tkiva. V dveh

dimenzijah odziv tkiva ponazorimo z modeliranjem celic z Voronojevim diagramom in izračunom vsiljene

transmembranske napetosti znotraj vsakega poligona oz. celice (Joshi et al. 2008). Slabost zgornjih

modelov je, da so v dveh dimenzijah, kjer je obnašanje celic drugačno kot v tridimenzionalnih pogojih in

vivo. V tkivu so celice namreč gosto skupaj, so povezane, pride do zastiranja električnega polja in nižanja

vsiljene transmembranske napetosti (Susil et al. 1998; Pucihar et al. 2007; Pavlin in Miklavčič 2009),

prisotna sta krvni obtok in imunski sistem. Povezavo med celicami in tkivom lahko ponazorimo s

tridimenzionalnimi numeričnimi modeli mrež okroglih celic (Pavlin et al. 2002; Ramos 2010). Simulacije

so bile narejene za enako velike, popolnoma okrogle, pravilno razporejene in električno nepovezane celice,

kar ni v skladu z dejanskimi razmerami v tkivu (Pucihar in Miklavčič 2009), obenem pa je bil največji

prostorninski delež celic omejen na vrednosti, nižje od tistih v tkivih. Z numeričnim modeliranjem

tridimenzionalnih okroglih različno velikih celic, naključno razporejenih po prostoru, lahko opišemo

spremembo električne prevodnosti celične membrane in delež prepustnih celic (Essone Mezeme et al.

2012b). S tridimenzionalnim modelom skupka posameznih celic različnih oblik lahko opišemo obnašanje

lista špinače pri elektroporaciji (Dymek et al. 2015). V literaturi (Huclova et al. 2010; Huclova et al. 2011;

Huclova et al. 2012) so celice in tkivo tudi modelirali v treh dimenzijah, model pa je opisal celice različnih

oblik, velikosti in prostorninskih deležev. Lastnosti posameznih celic so nato posplošili na tkiva. V okviru

doktorske disertacije smo model nadgradili z elektroporacijo, ki je že v nelinearnem območju odziva celic

(Dermol-Černe in Miklavčič 2018).

14

Razširjen povzetek v slovenskem jeziku - Namen

Namen Namen doktorske disertacije lahko povzamemo v dveh točkah:

1. optimizacija (fenomenoloških) matematičnih modelov, ki opisujejo prepustnost celične membrane,

depolarizacijo in celično smrt;

2. vključitev mehanizmov elektroporacije v modele sprememb električnih lastnosti tkiv in transporta

v tkivih pri elektrokemoterapiji.

V prvi točki smo prilegali različne obstoječe matematične modele povišane prepustnosti celične membrane

in celične smrti na rezultate poskusov in vitro. Najprej smo se osredotočili na vprašanje, ali lahko iz deleža

celic s prepustnimi membranami v homogenem električnem polju napovemo delež celic s prepustnimi

membranami v nehomogenem električnem polju. Modelirali smo časovno dinamiko vnosa barvila v celice

in ovrednotili povezavo med celjenjem celične membrane in vnosom barvila. Nato smo prešli na modele

celične smrti. Naredili smo pregled obstoječih modelov celične smrti, pri čemer večina izhaja iz živilske

industrije. Obstoječe modele celične smrti smo nato prilegali na rezultate poskusov, ki smo jih pridobili pri

ireverzibilni elektroporaciji celic. Primerjali smo prepustnost in depolarizacijo vzdražnih in nevzdražnih

celic ter oba pojava modelirali.

V drugi točki smo se osredotočili na spremembe električnih lastnosti celic in tkiv ter povečanje vnosa snovi

v celice. Modelirali smo spremembe električnih lastnosti kože pri elektroporaciji z dolgimi

nizkonapetostnimi pulzi ali s kratkimi visokonapetostnimi pulzi. Ovrednotili smo prispevek nastanka in

širjenja lokalnih transportnih območij v roženi plasti in nastanka por v membranah celic spodnjih plasti

kože k povišanju električne prevodnosti kože. Preverili smo, ali lahko nastanek por na membranah celic v

spodnjih plasteh kože izračunamo z asimptotskim modelom nastanka por in ali lahko nastanek lokalnih

transportnih območij povežemo z obliko tokovnonapetostnih meritev. Modelirali smo transport cisplatina v

podkožne tumorje na miših. Poskusi na živalih so zamudni, dragi, zaradi etičnih pomislekov pa lahko

preizkusimo le omejeno število parametrov. Preverili smo, ali bi lahko rezultate poskusov in vitro uporabili

za napoved odziva tkiv ter možnosti kasnejše uporabe pri načrtovanju posegov elektrokemoterapije. Ocenili

smo, kako bližina celic, prisotnost celičnega matriksa in zmanjšana površina celične membrane, ki je na

voljo za transport, vplivajo na vnos cisplatina v celice.

15

Razširjen povzetek v slovenskem jeziku - Rezultati in razprava

Rezultati in razprava V tem podpoglavju so na kratko povzeti vsi rezultati, pridobljeni v okviru doktorske disertacije. Razdeljeni

so skladno s tremi izvirnimi znanstvenimi prispevki: 1. Matematično modeliranje prepustnosti celične

membrane in celične smrti; 2. Prehod modeliranja z nivoja ene same celice na nivo tkiv, pri čemer je

predstavljen realističen tridimenzionalen model kože nadgrajen z modelom elektroporacije vseh sestavnih

delov in bistvenih plasti in 3. Matematično modeliranje transporta molekul (kemoterapevtika) skozi celično

membrano na osnovi modela dvojne poroznosti. O rezultatih tudi razpravljam, na kratko pa so pri vsakem

prispevku opisane še metode.

Želimo si, da bi lahko celostno predstavili pojav elektroporacije ter z modeli opisali dogajanje na več

ravneh (lipidni dvosloj, celice, tkiva) pri več procesih (spremembi električnih lastnost, transportu). Začeli

smo z matematičnimi modeli prepustnosti celične membrane in celične smrti, časovne dinamike vnosa

barvila v celice in intenzivnostno-časovne krivulje vzdražnih tkiv. Modeli so sicer dobro opisali rezultate,

niso pa vključevali mehanizmov elektroporacije. Zato smo se namesto na fenomenološke modele

osredotočili na vključevanje mehanizmov elektroporacije v modele tkiv pri spremembi njihovih električnih

lastnosti ter pri transportu majhnih molekul v tkivu in skozi celične membrane po elektroporaciji.

Matematični modeli prepustnosti celične membrane, celične smrti in depolarizacije

V okviru doktorske disertacije smo se ukvarjali z matematičnimi modeli prepustnosti celične membrane

(Dermol 2014; Dermol in Miklavčič 2014; Dermol et al. 2016), celične smrti (Dermol in Miklavčič 2015;

Dermol in Miklavčič 2016) in depolarizacije (Dermol-Černe et al. 2018). V literaturi smo poiskali že

obstoječe matematične modele in jih prilegali na rezultate poskusov.

Matematični modeli prepustnosti celične membrane

Povišanje prepustnosti celične membrane smo modelirali s štirimi različnimi modeli. Simetrična sigmoida

(Puc et al. 2003) in hiperbolični tangens (Essone Mezeme et al. 2012b) sta bila v literaturi že uporabljena,

na podlagi primerne sigmoidne oblike pa smo dodali asimetrično sigmoido in Gompertzovo krivuljo (Laird

1964; Dermol in Miklavčič 2014). Matematične modele prepustnosti celične membrane smo optimizirali in

primerjali z rezultati poskusov. Poskuse smo izvajali na celični liniji CHO-K1 (jajčne celice samice

kitajskega hrčka). Uporabili smo dva tipa elektrod – paralelne žične (slika 3a) pri poskusih v homogenem

električnem polju in igelne (slika 3b) pri poskusih v nehomogenem električnem polju. Električno polje med

paralelnimi žičnimi elektrodami smo v približku izračunali kot dovedeno napetost, deljeno z razdaljo med

elektrodami (Sweeney et al. 2016), električno polje okoli igelnih elektrod pa smo izračunali numerično.

Pritrjene celice različnih gostot smo v rastnem mediju v prisotnosti fluorescentnega barvila propidijev jodid

izpostavili enemu 1 ms pulzu pri različnih napetostih. Pet minut po dovedenih električnih pulzih smo rastni

medij s propidijevim jodidom zamenjali z rastnim medijem brez propidijevega jodida, s tem zaustavili vnos

barvila v celice in zajeli slike pod mikroskopom.

16


Slika 3: a) Paralelne žične elektrode, med katerimi je električno polje približno homogeno in b) igelne

elektrode, okoli katerih smo električno polje izračunali numerično. Povzeto po (Dermol in Miklavčič 2014).

Najprej smo določili delež celic s prepustnimi membranami, izpostavljenih homogenemu električnemu

polju. Na rezultate poskusov smo z metodo najmanjših kvadratov prilegali štiri matematične modele

prepustnosti celične membrane v odvisnosti od električnega polja. Optimizirane modele smo vstavili v

numerični model igelnih elektrod in z numerično izračunanim električnim poljem okoli igelnih elektrod

določili deleže prepustnih celic v posameznih območjih (slika 4). Predvidene deleže prepustnih celic v

posameznih območjih smo primerjali s preštetimi deleži. Ugotovili smo, da se mora za pravilno napoved

deleža prepustnih celic gostota celic v nehomogenem in v homogenem električnem polju ujemati - gostota

celic mora biti torej enaka v poskusih, na katerih model optimiziramo, in v poskusih, kjer odziv

napovedujemo. Primer prostorsko odvisne verjetnosti za prepustnost celic okoli igelnih elektrod lahko

vidimo na sliki 5. Matematične modele prepustnosti celične membrane bi torej lahko uporabili v

načrtovanju posegov, kjer bi namesto električnega polja v tkivu predstavili verjetnost za prepustnost tkiva v

odvisnosti od geometrije elektrod in parametrov električnih pulzov.

17


Slika 4: a) Fazno-kontrastna in b) fluorescentna slika območja okoli igelnih elektrod, kjer so bile celice

izpostavljene nehomogenemu električnemu polju. Območje okoli elektrod smo primerjali z c in d)

numerično izračunanim električnim poljem in določili število prepustnih celic znotraj določenih območij

električnega polja. Povzeto po (Dermol in Miklavčič 2014).

Slika 5: Verjetnost za prepustnost (p) okoli igelnih elektrod pri izpostavitvi celice enemu 1 ms pulzu pri

napetosti 120 V. Prehod iz numerično izračunanega električnega polja v verjetnost za prepustnost celične

membrane je bil narejen z Gompertzovo krivuljo. Povzeto po (Dermol in Miklavčič 2014).

V drugi študiji (Dermol et al. 2016) smo se ukvarjali z modeliranjem vnosa barvila v celice v odvisnosti od

časa. Osredotočili smo se na pojav povišanja občutljivosti celic na električne pulze (angl.

electrosensitization). V literaturi lahko zasledimo opažanja, da obstaja razlika v celični smrti, če dovedemo

električne pulze v enem vlaku ali če vlak razdelimo na več delov, med njimi pa počakamo več minut

(Pakhomova et al. 2011; Pakhomova et al. 2013a; Jiang et al. 2014; Jiang et al. 2015b; Muratori et al. 2016;

Muratori et al. 2017). Shema dovajanja pulzov v obliki enotnega ali razdeljenega vlaka je prikazana na sliki

6. Premor med posameznimi vlaki pulzov je po poročilih v literaturi bistveno povišal celično smrt pri enaki

dovedeni energiji. S tem se je povišala učinkovitost zdravljenja, kar bi lahko imelo pozitivne učinke tudi pri

medicinskih posegih z elektroporacijo.

18


Slika 6: Shema dovajanja pulzov. Levo je enotni vlak, kjer so bili naenkrat dovedeni vsi pulzi. Desno je

razdeljeni vlak, kjer so bili najprej dovedeni štirje pulzi, po 1 do 3 minutnem premoru pa še štirje pulzi.

Povzeto po (Dermol et al. 2016).

Preverjali smo, če je pojav povišanja občutljivosti celic odvisen tudi od sestave elektroporacijskega pufra,

t. j. tekočine, v kateri so elektroporirane celice. Poskuse smo izvajali na suspenziji celic CHO-K1. Pod

mikroskopom smo zajemali slike pred dovedenimi električnimi pulzi in 8 minut za njimi. Uporabili smo

paralelne žične elektrode, prikazane na sliki 3a. Celice smo električnim pulzom izpostavili v štirih različnih

elektroporacijskih pufrih, ki so se razlikovali v električni prevodnosti, osmolalnosti, koncentraciji saharoze

in kalcija. Dovedli smo enotni ali razdeljeni vlak električnih pulzov ter analizirali prispevek prvega in

drugega vlaka električnih pulzov. Časovno dinamiko smo modelirali s fenomenološkim modelom prvega

reda. Časovni potek vnosa barvila je shematično prikazan na sliki 7, kjer so napisane tudi enačbe, ki smo jih

uporabljali za opis dinamike vnosa barvila v celice, obenem pa so na sliki razloženi tudi parametri enačb.

Ugotovili smo, da lahko iz rezultatov vnosa barvil v molekule sklepamo tudi na celjenje celične membrane.

S pomočjo merjenja časovne dinamike vstopa fluorescentnega barvila propidijev jodid smo ugotovili, da

ima pufer, v katerem celice izpostavimo električnim pulzom, velik vpliv. Sklepali smo, da je povišanje

občutljivosti celic posledica skupnega delovanja več procesov ter ni splošno prisoten pojav. Za točno

določitev mehanizmov povišanja občutljivosti celic so potrebne nadaljnje študije.

19


Slika 7: Dinamika zapiranja por je bila modelirana s kinetičnim modelom prvega reda. Na sliki lahko

vidimo, kako sta bila posebej analizirana prispevek prvega in drugega vlaka pulzov. Povzeto po (Dermol et

al. 2016).

Matematični modeli preživetja celic

Ukvarjali smo se tudi z matematičnimi modeli celične smrti. Matematičnih modelov celične smrti poznamo

veliko, saj jih rutinsko uporabljajo za napoved učinkovitosti pasterizacije in sterilizacije v živilski industriji.

Pri načrtovanju medicinskih posegov elektroporacije matematični modeli niso razširjeni – trenutno

uporabljamo deterministično kritično vrednost za določanje meje med uničenim in nedotaknjenim tkivom.

Modele smo prilegali na rezultate poskusov. Naredili smo poskuse na celični suspenziji CHO-K1 v nizko

prevodnem kalijevem pufru. Kapljico celične suspenzije smo prenesli med jeklene paralelne elektrode

(slika 8) in jih izpostavili električnim pulzom podobnih parametrov, kot jih uporabljajo pri zdravljenju tkiv

z ireverzibilno elektroporacijo (do 90 pulzov dolžine 100 μs, dovedenih s ponavljalno frekvenco 1 Hz pri

električnih poljih do 4 kV/cm). Celično smrt smo ovrednotili s testom klonogenosti (Franken et al. 2006),

kjer smo po elektroporaciji v petrijevke nasadili celice v nizki gostoti. V šestih dneh so celice ustvarile

kolonije, ki smo jih nato fiksirali, obarvali, prešteli ter določili delež preživelih celic.

Slika 8: Jeklene paralelne elektrode, ki so bile uporabljene v poskusih. Med elektrode smo prenesli kapljico

celične suspenzije ter jo izpostavili električnim pulzom.

Ovrednotili smo več modelov celične smrti – kinetični model prvega reda, Hülshgerjev, Peleg-Fermijev,

Weibullov, logistični, prilagojen Gompertzov in Geeraerdov model. Ugotovili smo, da je najbolj primeren

Peleg-Fermijev model, ki je dobro opisal rezultate poskusov in vključeval dve neodvisni spremenljivki –

električno polje in število pulzov. V nasprotju z njim so ostali modeli vključevali le eno neodvisno

spremenljivko – ali čas izpostavitve (število pulzov zmnoženo z njihovim trajanjem) ali električno polje

med elektrodami. Na sliki 9 lahko vidimo rezultate poskusov in modele, ki smo jih prilegali na te podatke.

Vidimo lahko, da se uporabljeni modeli različno približajo rezultatom poskusov, parameter R2, napisan

poleg modela, pa nam pove še, kako dober je model v opisu podatkov. Bolj kot se vrednost R2 približa 1,

boljše je prileganje.

20


Slika 9: Modeli celične smrti (Weibullov, Peleg-Fermijev, prilagojen Gompertzov, logistični, kinetični

model prvega reda, Geeraerdov), s katerimi smo opisali rezultate poskusov celične smrti. Na sliki a so

modeli v odvisnosti od dovedenega električnega polja, na sliki b pa od zmnožka števila pulzov in njihovega

trajanja. Povzeto po (Dermol in Miklavčič 2015).

Ugotovili smo, da bi bili matematični modeli prepustnosti celične membrane in celične smrti lahko

uporabni v načrtovanju posegov. Dobra lastnost teh modelov je, da jih hitro prilegamo na rezultate

poskusov, vrednosti med točkami lahko interpoliramo. Slabost modelov je, da jih je potrebno optimizirati

za vsak tip celic posebej, prenosljivost na tkiva pa je vprašljiva.

Modeliranje depolarizacije celic

Pri načrtovanju medicinskih posegov z elektroporacijo je smiselno upoštevali tudi vzdražnost tkiv. V

primeru proženja akcijskega potenciala mišic in živcev pride do neželenega krčenja mišic in bolečine.

Ugotavljali smo: 1. ali se vzdražna in nevzdražna tkiva na elektroporacijo odzivajo enako, 2. ali lahko z

elektroporacijo zdravimo tumorje osrednjega živčnega sistema in 3. ali so lastnosti vzdražnih tkiv po

elektroporaciji spremenjene. Več študij je sicer pokazalo, da so bili učinki elektroporacije na vzdražna tkiva

le kratkoročni – tkiva so si opomogla fiziološko, histološko in funkcionalno (Onik et al. 2007; Li et al.

2011; Schoellnast et al. 2011; Jiang et al. 2014; Tschon et al. 2015; Casciola et al. 2017).

S poskusi in modeliranjem smo ovrednotili depolarizacijo in elektroporacijo štirih različnih celičnih linij –

nevzdražne celične linije CHO-K1, nevzdražne celične linije U-87 MG (človeški glioblastom), nevzdražne

nediferencirane celične linije HT22 (mišji nevroni iz hipokampusa) in vzdražne diferencirane celične linije

HT22. Depolarizacija pomeni povišanje transmembranske napetosti, do katere lahko pride tako na

vzdražnih kot na nevzdražnih celicah, ko jih izpostavimo zunanjemu električnemu polju. Če presežemo

pragovno vrednost transmembranske napetosti, pri vzdražnih celicah pride do proženja akcijskega

potenciala.

Najprej smo s poskusi določili depolarizacijsko (za nevzdražne) oz. vzdražnostno (za vzdražne celice)

intenzivnostno-časovno krivuljo za en pulz dolžine od 10 ns do 10 ms. Za določanje depolarizacije smo

uporabljali fluorescentno barvilo (Baxter et al. 2002). Barvilo je v začetku v zunajceličnem prostoru, po

povišanju transmembranske napetosti pa lahko preide v notranjost celice in povzroči v njej dvig signala.

21


Intenzivnostno-časovno krivuljo vzdražnih celic smo opisali z dvema modeloma – s fenomenološko

Lapicquevo krivuljo (Boinagrov et al. 2010) in Hodgkin-Huxleyjevim modelom (Santamaria in Bower

2009; Boinagrov et al. 2010). Hodgkin-Huxleyjev model je sistem diferencialnih enačb, ki opisuje

dinamiko odpiranja in zapiranja napetostno odvisnih kanalov. Akcijski potencial, modeliran s Hodgkin-

Huxleyjevim modelom, lahko vidimo na sliki 10. Ugotovili smo, da je primernejši Hodgkin-Huxleyjev

model, čeprav bi morali za najboljši opis naših rezultatov njegove parametre optimizirati, kar bi bilo zaradi

njihovega visokega števila težavno.

Intenzivnostno-časovno krivuljo (slika 11) smo primerjali z rezultati poskusov elektroporacije po enem

pulzu različnih dolžin (Pucihar et al. 2011). V isti študiji so rezultate poskusov modelirali s Saulisovim

modelom (Pucihar et al. 2011). Dodali smo še modeliranje s časovno odvisno Schwannovo enačbo za

eliptične celice v električnem polju (Kotnik et al. 1998; Kotnik in Miklavčič 2000b; Valič et al. 2003). Pri

Schwannovi enačbi smo predpostavljali, da je kritična vsiljena transmembranska napetost enaka pri vseh

dolžinah pulzov, kar ni nujno pravilno. Ugotovili smo, da elektroporacijo boljše opiše Saulisov model.

Slika 10: Primer depolarizacije z akcijskim potencialom in brez njega pri modeliranju s Hodgkin-

Huxleyjevim modelom (Boinagrov et al. 2010). Modre in oranžne črte opisujejo napetost na katodnem in na

anodnem polu celice. Po dovedenem pulzu se vidi le še oranžna črta, saj se z modro prekrivata. V primeru

na sliki smo simulirali en 1 ms pulz tik pod pragom za proženje akcijskega potenciala in nad njim.

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Slika 11: Intenzivnostno-časovna krivulja CHO, U-87 MG, nediferenciranih in diferenciranih HT22 celic.

Intenzivnostno-časovno krivuljo smo modelirali z Lapicqueovo krivuljo in s Hodgkin-Huxleyjevim

modelom. Prikazana je tudi prepustnostna krivulja CHO celic, pridobljena v okviru študije (Pucihar et al.

2011) in modelirana s Saulisovim modelom in Schwannovo enačbo. Črne črte in y-os kažejo prepustnost

celične membrane po enem pulzu. Sive črte in desna y-os kažejo depolarizacijo oz. akcijski potencial po

enem pulzu.

Zanimala nas je tudi elektroporacija nevzdražnih in vzdražnih celičnih linij, zato smo vse štiri celične linije

izpostavili pulzom podobnih parametrov, kot se običajno uporabljajo pri elektrokemoterapiji – osem pulzov

dolžine 100 μs, dovedenih s ponavljalno frekvenco 1 Hz. Vnos smo določali s fluorescentnim barvilom Yo-

Pro-1®. Merili smo časovno dinamiko vnosa barvila v celice, kot lahko vidimo na sliki 12. Iz časovne

dinamike smo nato določili prepustnostno krivuljo, ki je prikazana na sliki 13a. S poskusi določene

prepustnostne krivulje smo modelirali s simetrično sigmoido (Dermol in Miklavčič 2014), kar je prikazano

na sliki 13b.

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Slika 12: Časovna dinamika vnosa barvila Yo-Pro-1® v celice štirih celičnih linij. Dinamiko smo lahko

opisali z modelom prvega reda.

Slika 13: Povišanje prepustnosti membran štirih celičnih linij po dovedenih pulzih podobnih parametrov,

kot se uporabljajo pri elektrokemoterapiji. S poskusi določene točke so na sliki a, optimizirana simetrična

sigmoida pa na sliki b.

Ugotovili smo, da je prag za depolarizacijo nevzdražnih celic nižji kot prag za depolarizacijo vzdražnih

celic, kar lahko pojasnimo s tem, da imajo diferencirane celice, ki se ne delijo, nižjo mirovno

transmembransko napetost (Levin in Stevenson 2012). Vse štiri celične linije so se podobno odzvale na

elektroporacijo s pulzi podobnih parametrov, kot so uporabljeni pri elektrokemoterapiji. To nakazuje, da je

elektroporacija primerna za zdravljenje vzdražnih tkiv. Naši rezultati kažejo, da je možno optimizirati

parametre električnih pulzov tako, da dobimo čim višji vnos molekul v celice, obenem pa se izognemo

proženju akcijskega potenciala. V prihodnosti bo potrebno določiti, če depolarizacija in vitro in vzdražnost

korelirata s krčenjem mišic in bolečino in vivo.

Prehod modeliranja z ravni ene same celice na raven tkiv z upoštevanjem njegove strukture

Zaradi elektroporacije se spremenijo električne lastnosti celic in posledično tudi tkiv. Zgradili smo model

kože, ki temelji na geometriji celic (keratinocitov, korneocitov) in lahko opiše spremembe električnih

lastnosti kože med elektroporacijo. Elektroporacijo smo modelirali s spremembo prevodnosti kože z

nastankom lokalnih transportnih območij v roženi plasti ter nastankom por na membranah celic spodnjih

plasti kože.

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Model je temeljil na rezultatih poskusov, pridobljenih v okviru že objavljene študije (Zorec et al. 2013a).

Na kratko: v Franzevih difuzijskih celicah so pulzom izpostavili 350 μm debele kose prašičje kože z ušes.

Elektrode so bile postavljene 0,2 cm nad kožo in 0,5 cm pod njo. Dovedli so dva različna protokola: dolge

nizkonapetostne pulze (3x45 V dolžine 250 ms in 100 ms pavze med njimi) ali kratke visokonapetostne

pulze (3x500 V dolžine 500 μs, 500 μs pavze med njimi). 350 μm debela koža je glede na model (Huclova

et al. 2012) sestavljena iz štirih slojev – rožene plasti, epidermisa, papilarnega dermisa in žilne plasti.

Naš model smo osnovali na že obstoječem modelu (Huclova et al. 2012), kjer so lastnosti posameznih

slojev kože izračunane s pomočjo električnih in geometrijskih lastnosti posameznih celic. Tako kot v študiji

(Huclova et al. 2012) smo električne lastnosti slojev z nizkim prostorninskim deležem celic izračunali s

Hanai-Bruggemanovo formulo, ki sicer spada med t. i. enačbe mešanja. Lastnosti slojev z višjim

prostorninskim deležem celic (nad 80%) smo izračunali z modelom enotske celice. Enotska celica je s

kvadrom zunajcelične tekočine obdana biološka celica. Tri enotske celice, ki smo jih zgradili v okviru naše

študije, so prikazane na sliki 14. Oblika biološke celice je bila opisana s pomočjo t. i. 'superformule' (Gielis

2003). Biološki celici smo pripisali ustrezne električne lastnosti. Z velikostjo enotske celice smo spreminjali

prostorninski delež celic v tkivu. Enotsko celico smo izpostavili sinusni napetosti različnih frekvenc v vseh

treh smereh (x, y, z) in preko admitance izračunali frekvenčno odvisen kompleksni tenzor električnih

lastnosti enotske celice (𝜀𝜀*). Posameznim plastem v homogenemu modelu kože smo pripisali izračunane

električne lastnosti enotske celice in numerične izračune poenostavili na raven homogenih modelov tkiva. V

primeru rožene plasti smo električne lastnosti izračunali preko korneocitov v enotski celici, pri epidermisu

pa preko keratinocitov v enotski celici. Z izbranim modelom lahko modeliramo realistične oblike celic v

visokih prostorninskih deležih, anizotropijo in heterogenost tkiv. V naši študiji smo v model vključili

spremembe električnih lastnosti kože med elektroporacijo in model primerjali s tokovnonapetostnimi

meritvami.

Slika 14: Enotske celice, s katerimi smo predstavili (a) keratinocit, iz katerega smo nato izračunali lastnosti

rožene plasti, (b) korneocit, iz katerega smo izračunali lastnosti epidermisa in (c) krogle lipidov v

zunajcelični tekočini, iz katerih smo nato izračunali lastnosti papilarnega dermisa. Geometrija keratinocita

in korneocita je bila zgrajena s 'superformulo' (Gielis 2003; Huclova et al. 2012), zaradi simetrije pa smo

modelirali le eno osmino celice. Geometrija papilarnega dermisa so krogle v ploskovno centrirani mreži.

Povzeto po (Dermol-Černe in Miklavčič 2018).

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Pri elektroporaciji kože pride do dveh pojavov – v roženi plasti nastanejo lokalna transportna območja, na

membranah celic spodnjih plasti pa pride do elektroporacije oz. nastanka por. Pri elektroporaciji kože z

dolgimi nizkonapetostnimi pulzi se spreminja velikost lokalnih transportnih območij, saj okoli njih prihaja

do topljenja lipidov zaradi Joulovega gretja. Spremembo velikosti lokalnih transportnih območij med

nizkonapetostnimi pulzi smo prevzeli iz literature (Zorec et al. 2013a). Pri elektroporaciji s kratkimi

visokonapetostnimi pulzi se poviša gostota lokalnih transportnih območij, njihova velikost pa se ne

spreminja. Spremembo gostote lokalnih transportnih območij smo določili na podlagi najboljšega ujemanja

s poskusi. Obenem smo z asimptotsko enačbo (Neu in Krassowska 1999; Neu in Neu 2009) izračunali

gostoto por na membranah celic spodnjih plasti kože, pri čemer smo predpostavljali, da padec napetosti na

koži lahko določimo, če kožo razumemo kot napetostni delilnik. Ekvivalentno vezje Franzeve difuzijske

celice in kože v njej je prikazano na sliki 15.

Slika 15: Shema, ekvivalentno vezje in fotografija Franzeve difuzijske celice. Na sliki (a) lahko vidimo, da

je bila koža na obeh straneh obdana z elektroporacijskim pufrom, v katerega so bile vstavljene platinaste

elektrode. V (b) ekvivalentnem vezju so z upori so predstavljeni upornost stika kovina-elektrolit (Relektrode),

elektroporacijski pufer (RPBS) in posamezne plasti kože (RSC – rožena plast, RE – epidermis, RPD – papilarni

dermis, RUVP – žilna plast). Na sliki (c) lahko vidimo fotografijo Franzeve celice.

Spremenjene električne lastnosti posameznih tkiv smo nato vstavili v numerični model Franzeve difuzijske

celice ter modelirali tok skozi kožo in napetost na njej, če na elektrode dovedemo napetost, izmerjeno med

poskusom na izhodu pulznega generatorja. Spreminjali smo gostoto transportnih območij, električno

prevodnost transportnih območij in električno prevodnost spodnjih plasti kože, dokler nismo dosegli

dobrega ujemanja s poskusi. Določili smo (lokalni) optimum parametrov. Zaradi visokega števila

parametrov, ki jih v modelu lahko spreminjamo, in pomanjkanja meritev pri elektroporaciji kože,

globalnega optimuma nismo mogli določiti. Za točno določitev vrednosti parametrov bi potrebovali več

meritev, npr. električne lastnosti posameznih slojev pred elektroporacijo in po njej ter električno prevodnost

lokalnih transportnih območij.

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Ugotovili smo, da lahko dobro opišemo elektroporacijo z dolgimi nizkonapetostnimi pulzi, če upoštevamo

širitev lokalnih transportnih območij, spremembo električne prevodnosti spodnjih plasti in polarizacijo

elektrod. Izmerjeni in modelirani tok in napetost lahko vidimo na slikah 16a in 16b. Ujemanje modela in

meritev pri elektroporaciji s kratkimi visokonapetostnimi pulzi je bilo slabše – napetost se je sicer dobro

ujemala, a je model tok podcenjeval za približno 50%, kar lahko vidimo na slikah 16c in 16d. Razlog bi

lahko bil v elektrokemijskih reakcijah, ki se dogajajo med elektroporacijo. Kemija pri visokonapetostnih

pulzih je relativno neraziskano področje, obstaja le nekaj študij, ki se ukvarjajo z izpustom kovin iz elektrod

(Kotnik et al. 2001; Roodenburg et al. 2005; Pataro et al. 2014) in z možnimi kemijskimi reakcijami, ki se

dogajajo na elektrodah ali ob njih (Pataro et al. 2015a; Pataro et al. 2015b). Le redki modeli elektroporacije

vključujejo elektrokemijo, npr. polarizacijo elektrod (Šel et al. 2003), čeprav je, kot se je izkazalo pri našem

modelu, polarizacija elektrod bistveno zmanjšala napetost, dovedeno na kožo.

Slika 16: Z modelom električnih lastnosti elektroporirane kože smo opisali tokovnonapetostne meritve pri

dovajanju dolgih nizkonapetostnih pulzov in kratkih visokonapetostnih pulzov. Pri nizkonapetostnih pulzih

smo dobro opisali meritve, pri visokonapetostnih pa je model podcenjeval izmerjeni električni tok. Pri

visokonapetostnih pulzih so bili prisotni še drugi mehanizmi, kot so npr. elektrokemijske reakcije, ki jih v

model nismo vključili. Povzeto po (Dermol-Černe in Miklavčič 2018).

Naš model povezuje procese, ki se med elektroporacijo dogajajo na posamezni celici (nastanek por v celični

membrani) in na plasti kože (nastanek lokalnih transportnih območij v roženi plasti), s tkivi (plasti kože) in

celo z organi (koža). Z modelom lahko opišemo spremembe električnih lastnosti tkiv, pri čemer izhajamo iz

mikrostrukture tkiva, obenem pa ostajajo numerični izračuni relativno hitri in preprosti. V prihodnosti bi v

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model lahko vključili še termične pojave ali elektrokemijo. S tem lahko opišemo katero koli tkivo, če

poznamo električne in geometrijske lastnosti osnovnih gradnikov tkiva.

Modeliranje transporta molekul skozi celično membrano

Pri zdravljenju tumorjev z elektrokemoterapijo je pomembno, da je v celotnem tumorju vzpostavljeno

dovolj visoko električno polje (Miklavčič et al. 2006a), obenem pa mora biti v celicah dovolj molekul

kemoterapevtika, da zagotovimo celično smrt (Tounekti et al. 1993; Tounekti et al. 2001; Miklavčič et al.

2014). Pri načrtovanju posegov bi bilo torej smiselno modelirati tudi transport kemoterapevtika v tumorske

celice. Modelirali smo transport cisplatina v tumorske celice pri elektrokemoterapiji tumorjev. Za to smo

uporabili model dvojne poroznosti. Za postavitev modela smo izvedli poskuse in vitro, v sodelovanju z

Onkološkim inštitutom pa tudi poskuse in vivo. Meritve mase platine so potekale v sodelovanju z

Inštitutom Jožef Štefan.

V študiji transporta majhnih molekul smo rezultate vnosa cisplatina v celice in vitro uporabili za opis

transporta cisplatina v tumorjih in vivo. Celično suspenzijo mišjih melanomskih celic B16-F1 smo v

rastnem mediju s serumom izpostavili 100 μs dolgim električnim pulzom s ponavljalno frekvenco 1 Hz. Z

induktivno sklopljeno plazmo z masnim spektrometrom smo izmerili znotrajcelično maso platine. Najprej

smo izvedli poskuse, pri katerih smo optimizirali protokol elektroporacije in vitro. Celice smo

elektroporirali pri različnih zunajceličnih koncentracijah cisplatina in ugotovili, da je znotrajcelična masa

platine linearno odvisna od zunajcelične koncentracije cisplatina (slika 17a). Zato smo nadaljnje poskuse

izvedli pri 100 μM zunajcelični koncentraciji cisplatina, kar je dovolj visoka koncentracija, da smo lahko

pri meritvah mase platine razlikovali med pulzi podobnih parametrov. Izmerili smo tudi dinamiko vnosa

cisplatina v celice pri 100 μM zunajceličnem cisplatinu in ugotovili, da že 5 minut po pulzih dosežemo

plato, zato smo vse nadaljnje poskuse izvajali z 10 minutno inkubacijo (slika 17b). Cisplatin se lahko veže

na proteine v serumu in s tem postane biološko neaktiven (Takahashi et al. 1985). Celice smo

elektroporirali s serumom ali brez njega in ugotovili, da v naših poskusih serum ni vplival na vnos

cisplatina (slika 17c). Izmerili smo tudi celjenje membrane, tako da smo cisplatin dodali do 20 minut po

pulzih. Celjenje por smo opisali z dinamiko prvega reda in izračunali, da je bila časovna konstanta celjenja

por 2,29 minute. Po optimizaciji osnovnega protokola smo spreminjali število pulzov od 1 do 64 in

električno polje od 0,4 kV/cm do 1,2 kV/cm. Maso znotrajcelične platine pri teh parametrih lahko vidimo

na sliki 18.

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Slika 17: Rezultati poskusov in vitro elektroporacije celic s cisplatinom. a) Znotrajcelična masa platine je

bila linearno odvisna od zunajcelične koncentracije cisplatina. b) Vpliv časa inkubacije na znotrajcelično

maso platine. Plato smo dosegli že 5 minut po dovedenih pulzih. c) Vpliv seruma na znotrajcelično maso

platine. Ugotovili smo, da v 10 minutah po pulzih serum ne vpliva na vnos cisplatina. d) Po dovedenih

električnih pulzih se je celična membrana celila z dinamiko prvega reda.

Z modelom dvojne poroznosti (Mahnič-Kalamiza et al. 2014a; Mahnič-Kalamiza in Vorobiev 2014;

Mahnič-Kalamiza et al. 2015; Mahnič-Kalamiza et al. 2017) smo izračunali koeficient prepustnosti celične

membrane v odvisnosti od parametrov električnega polja. Obenem smo naredili poskuse in vivo, kjer smo v

podkožni tumor B16-F10 intratumorsko vbrizgali cisplatin ter izmerili maso platine v serumu, celicah in

intersticijski frakciji tik pred dovajanjem pulzov ter eno uro po končani terapiji. Zgradili smo numerični

model tumorja ter z in vitro izračunanim koeficientom prepustnosti in difuzijsko konstanto cisplatina v

intersticijski frakciji tumorja opisali maso platine v tumorjih in vivo. Opis je bil mogoč, če smo transport

cisplatina po intersticijski frakciji zmanjšali s pomočjo transformacijskega koeficienta, s katerim smo

upoštevali razlike v transportu in vitro ter in vivo. Pri transportu in vivo je zaradi bližine celic, visokega

prostorninskega deleža celic ter prisotnosti celičnega matriksa na voljo manjša površina membrane, skozi

katero lahko v celice pride cisplatin, otežena pa je tudi difuzija molekul po intersticijski frakciji. Obenem je

zaradi visokega prostorninskega deleža in bližine celic zmanjšana tudi vsiljena transmembranska napetost

na celicah. Prostorsko porazdelitev cisplatina eno uro po dovedenih pulzih, kar ustreza časovni točki, ko je

bil tumor in vivo izrezan, lahko vidimo na sliki 19. Očitno je, da je prostorska porazdelitev koncentracije

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cisplatina v tumorju nehomogena in se niža, ko se bližamo zunanjemu robu tumorja. V prihodnosti je

potrebno še preveriti vpliv tipa tumorja in gostote celic v njem ter preveriti delovanje modela pri drugačnih

parametrih električnega polja. Naš model ponuja korak naprej pri modeliranju transporta pri elektroporaciji

na ravni tkiv, upoštevajoč transport na ravni ene celice. Z našim modelom bi lahko zmanjšali število

poskusov in vivo ter bolj točno napovedali učinkovitost zdravljenja pri načrtih medicinskih posegov z

elektroporacijo.

Slika 18: Izmerjene znotrajcelične mase platine pri elektroporaciji celične suspenzije celic mišjega

melanoma B16-F1. Vidimo lahko, da najvišji vnos dosežemo pri 8x100 μs pulzih pri 1,2 kV/cm.

Slika 19: Modelirana porazdelitev a) znotrajcelične in b) zunajcelične koncentracije cisplatina v tumorju v

mol/m3. Koncentracija obeh je najvišja v sredini tumorja, nato pa postopoma upada do roba tumorja, kjer

je koncentracija 0 mol/m3.

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Razširjen povzetek v slovenskem jeziku - Zaključek

Zaključek V doktorski disertaciji sem se ukvarjala z modeliranjem procesov pri elektroporaciji na ravni celic ter na

ravni tkiv. Na ravni celičnih suspenzij ali posameznih celic in vitro je poskuse relativno lahko izvajati,

preizkusimo lahko več parametrov brez etičnih pomislekov, ki se pojavijo pri izvajanju poskusov in vivo

(Workman et al. 2010). Naš cilj je bil, da z rezultati poskusov in vitro modeliramo dogajanje med

elektroporacijo in vivo in po njej . Modele, ki smo jih zgradili, bi lahko v prihodnosti vključili v načrtovanje

medicinskih posegov z elektroporacijo.

V prvem delu doktorske disertacije so opisani matematični modeli prepustnosti celične membrane in

celične smrti. Ugotovili smo, da lahko s temi modeli dobro opišemo rezultate poskusov in vitro. Naše

optimizirane modele celične smrti in vitro so že uporabili pri opisu ireverzibilne elektroporacije tkiv

(Kranjc et al. 2017) in dosegli dobro ujemanje. V modelih celične smrti se z napovedjo približujemo

preživetju 0, a ga nikoli ne dosežemo. Vprašanje je, kako nizko preživetje je potrebno zagotoviti, da

uničimo tumor v celoti. Pri elektrokemoterapiji in pri netermičnem odstranjevanju tkiva z elektroporacijo

preostale celice uniči imunski sistem (Neal et al. 2013; Serša et al. 2015).

Analizirali smo časovni potek vnosa barvila v celice pri elektroporaciji, kar lahko modeliramo z modelom

prvega reda, ki tudi ustreza dinamiki celjenja por v celični membrani. V več študijah so določali hitrost

celjenja celične membrane po elektroporaciji in izmerili, da je v območju minut tako in vitro kot tudi in

vivo (Kinosita in Tsong 1977; Saulis et al. 1991; Čemažar et al. 1998; Shirakashi et al. 2004; Kandušer et

al. 2006; Demiryurek et al. 2015), kar se sklada z našimi rezultati.

Pri elektroporaciji vzdražnih tkiv lahko pride do proženja akcijskega potenciala in krčenja mišic ter

bolečine. S poskusi smo določili intenzivnostno-časovno krivuljo vzdražnih in nevzdražnih celičnih linij ter

jo modelirali. V prihodnosti je potrebno ugotoviti, do kolikšne mere lahko iz in vitro depolarizacije in

akcijskega potenciala napovemo krčenje mišic in bolečino in vivo. Pri uporabi intenzivnostno-časovnih

krivulj za napovedovanje akcijskega potenciala v tkivih je potrebno upoštevati tudi, da nekaj časa po

dovedenih pulzih tkivo ne zmore prožiti akcijskega potenciala. Izkazalo se je, da lahko elektroporacijski

pulzi povzročijo škodo na membranskih proteinih, kot so napetostno odvisni kanali v celični membrani

(Chen et al. 2006b; Azan et al. 2017). Čas odziva napetostno odvisnih kanalov je v okvirju milisekund

(Gadsby 2009), kar je bistveno daljše kot nanosekundni pulzi. Vprašanje je torej, kako je sploh možno, da z

nanosekundnimi pulzi prožimo akcijski potencial vzdražnih celic (Pakhomov et al. 2017).

V okviru študije o vzdražnih tkivih smo opazili, da pri pulzih dolžine 1 μs depolarizacijska intenzivnostno-

časovna krivulja in krivulja prepustnosti po y-osi prideta zelo blizu skupaj. To opažanje je pomembno pri

razlagi, zakaj s kratkimi bipolarnimi pulzi povzročimo prepustnost tkiv, obenem pa ne pride do vzdraženja.

Pri 1 μs dosežemo namreč plato pri obeh intenzivnostno-časovnih krivuljah. Do podobnega zaključka so

prišli v literaturi (Mercadal et al. 2017). Naši rezultati nakazujejo, da bi namesto s kratkimi bipolarnimi

pulzi enak učinek dosegli s kratkimi monopolarnimi pulzi.

V drugem delu doktorske disertacije smo modelirali spremembe električnih lastnosti kože med

elektroporacijo z nastankom por v celičnih membranah ter z nastankom lokalnih transportnih območij v

31


roženi plasti. V prihodnosti je potrebno podoben model zgraditi še za druga tkiva (npr. jetra ali mišice) in ga

primerjati s tokovnonapetostnimi meritvami med elektroporacijo. Nadgradnja modela bi lahko bila, da

namesto asimptotske oblike enačbe por uporabimo enačbe, ki predvidevajo spremembe v premeru por

(Smith et al. 2004), s čimer bi prevodnost posamezne plasti kože določili z mehanizmi elektroporacije in ne

le z najboljšim ujemanjem z meritvami. Z modelom smo sicer dosegli dobro ujemanje z meritvami pri

dolgih nizkonapetostnih pulzih, pri kratkih visokonapetostnih pulzih pa ne. Pri kratkih visokonapetostnih

pulzih se je pojavila dodatna komponenta električnega toka, ki je nismo mogli pojasniti z nastankom

lokalnih transportnih območij. Tudi če smo predpostavili, da je cela rožena plast eno samo veliko

transportno območje, je bil tok skozi kožo prenizek, obenem pa je bil v tem primeru padec napetosti na koži

zelo majhen zaradi nizke upornosti kože. Pri meritvah napetosti na koži so uporabljali tanke bakrene žice.

Na vsako stran kože je bila dana ena žica, pri tem se je ustvarila visoko prevodna pot za električni tok, saj je

baker bolj prevoden kot koža ali pufer, v katerem je bila koža. S tem bi se lahko električni tok lokalno

povišal, obenem pa ne bi vplival na padec napetosti na celotni koži. Vsekakor je v prihodnosti potrebno

ovrednotiti elektrokemijske reakcije, ki se odvijajo med dovajanjem visokonapetostnih pulzov. Vemo

namreč, da pride do izhajanja kovin iz elektrod, spremeni se pH tekočine, opazen je nastanek mehurčkov, ki

so lahko ali posledica lokalnega izparevanja vode ali kemijskih reakcij, kjer nastajajo plini.

V model kože bi lahko vključili tudi termične vplive električnih pulzov. Zaradi Joulovega gretja se namreč

poviša temperatura, z njo pa tudi električna prevodnost. Gretje je bilo v našem modelu vključeno posredno,

z večanjem premera lokalnih transportnih območij, kar temelji na taljenju lipidov v roženi plasti. Morda bi

v prihodnosti lahko v naš model vključili še model taljenja lipidov in nastanka lokalnih transportnih

območij (Becker in Kuznetsov 2007; Becker 2012; Becker et al. 2014) – v našem modelu smo namreč

velikosti lokalnih transportnih območij privzeli iz literature (Zorec et al. 2013a).

V modelu smo vsiljeno transmembransko napetost na celicah v spodnjih plasteh kože ocenili tako, da smo

kožo obravnavali kot napetostni delilnik. Upoštevali smo debelino posamezne plasti in geometrijo celice.

Pri tem smo zanemarili vpliv bližine celic, ki zmanjšuje vsiljeno transmembransko napetost. Možno je, da

smo precenili spremembo v električni prevodnosti spodnjih plasti kože, obenem pa podcenili prevodnost

lokalnih transportnih območij, njihovo gostoto in velikost. Električna prevodnost posameznih plasti kože

vpliva na porazdelitev električnega polja in obratno – s pomočjo te odvisnosti bi v model lahko vključili

tudi napake.

V tretjem delu doktorske disertacije smo se ukvarjali s transportom snovi po tkivih pri elektroporaciji. Naš

model je veljaven za majhne molekule, podobne velikosti kot cisplatin (300 g/mol), kot so propidijev jodid,

bleomicin, lucifer rumeno. Pokazalo se je, da večje molekule težje vstopajo preko celične membrane kot

majhne molekule pri enakih parametrih električnih pulzov (Maček Lebar et al. 1998) že zaradi bistveno

manjše difuzijske konstante večjih molekul po intersticijski frakciji tumorja. Pokazalo se je tudi, da večje

molekule v celice vstopajo s pomočjo elektroforeze, manjše pa s pomočjo difuzije. Večje molekule (DNK)

celično membrano prečkajo v več korakih, ki so opisani v literaturi (Rosazza et al. 2016). Predpostavili

smo, da je ves transport difuzijski, kar pa ni nujno pravilno. Zaradi povišanega intersticijskega tlaka v

tumorju (Baxter in Jain 1989; Boucher et al. 1990; Pušenjak in Miklavčič 2000) je povišan konvekcijski

transport molekul iz njega, kar pomeni, da kemoterapevtik v tumor težko vstopi, ko je doveden sistemsko,

32


oziroma da ga iz tumorja hitro odnaša, ko je doveden intratumorsko. Obstoječe meritve konvekcijske

hitrosti v tumorju se niso skladale z našimi meritvami mase platine v celicah pred pulzi in eno uro po njih –

izračunali smo namreč, da bi iz tumorja ves cisplatin izginil v nekaj minutah. V prihodnosti bi bilo smiselno

ovrednotiti konvekcijski transport v tumorjih, ki jih uporabljamo v okviru naše študije. Model smo trenutno

preizkusili le na eni vrsti tumorja. Znano je, da se tumorji bistveno razlikujejo v prostorninskem deležu

celic in matriksa (Mesojednik et al. 2007), kar seveda vpliva na difuzijsko konstanto molekul po tumorju ter

na površino celičnih membran, skozi katere lahko poteka transport. V prihodnosti bi lahko ovrednotili naš

model še na drugih tipih tumorjev in ugotovili, ali obstaja določena preslikava med morfologijo tumorja,

koeficientom transformacije in transportom po tumorju. Možno je, da takšna transformacija obstaja – višji

kot je prostorninski delež celic v tumorju, nižji je transport kemoterapevtika v celice in nižji je

transformacijski koeficient. Obenem se zastavlja vprašanje, če je transformacijski koeficient odvisen tudi od

parametrov električnega polja. Trenutno smo model preizkusili le pri osmih pulzih in eni dovedeni

napetosti. Smiselno bi bilo poskuse in vivo narediti še pri drugačnih parametrih električnih pulzov.

Pri intratumorskem dovajanju kemoterapevtika se izognemo običajni poti do celic v tumorju, ki jo mora

kemoterapevtik preiti, če ga dovedemo sistemsko. Po sistemski administraciji mora namreč iz krvnih žil

skozi žilne stene priti v okoliška tkiva (ali neposredno v tumorsko tkivo) in nato skozi intersticijsko frakcijo

tumorja, kjer transport ovira višji intersticijski tlak v tumorju kot v okoliškem tkivu (Jain 1999; Jang et al.

2003), prečkati celično membrano in priti v celice. Če bi naš model želeli uporabljati tudi pri sistemski

administraciji kemoterapevtika, bi morali dodati še izračun začetne koncentracije kemoterapevtika v

tumorju. Pri intratumorski injekciji smo namreč lahko utemeljeno predpostavili, da je v tumorju začetna

koncentracija kemoterapevtika enaka vbrizgani koncentraciji in je homogeno porazdeljena po celem

tumorju.

V poskusih in vitro je bil cisplatin ali v celicah ali v zunajcelični tekočini. V poskusih in vivo smo vbrizgali

80 μl cisplatina oz. 26 µg platine v 35 mm3 velike tumorje. Volumen injekcije je bil dvakrat večji kot

volumen tumorja, ker smo uporabili že obstoječe rezultate iz literature (Uršič et al. 2018). V poskusih in

vivo smo ugotovili, da je bilo 2 minuti po vbrizganju v tumorju in serumu le še 27% vbrizgane mase

platine. Del cisplatina smo izgubili zaradi manjšega volumna tumorja od injiciranega volumna in zaradi

izpiranja cisplatina iz tumorja. Če smo tumor izrezali eno uro po terapiji, je bilo vbrizgane mase platine v

tumorju in v serumu 2% v kontrolnih vzorcih in 7% v elektroporiranih vzorcih. Del cisplatina smo izgubili

pri mehanskem uničenju tumorja ter pri zbiranju krvi. Del cisplatina pa se je verjetno nabral v različnih

organih, kar bi kasneje lahko povzročilo stranske učinke (Sancho-Martínez et al. 2012). V prihodnosti bi

lahko ovrednotili, ali bi z vbrizganjem manjšega volumna cisplatina lahko ohranili enak učinek

elektroporacije, obenem pa bi zmanjšali stranske učinke. Opaženo je bilo, da volumen vbrizganega kalcija

ali virusov ni vplival na učinkovitost zdravljenja (Wang 2006; Frandsen et al. 2017). To lahko pojasnimo z

našimi poskusi, pri katerih smo določali začetno koncentracijo cisplatina v tumorju. Čeprav se je pri

vbrizganju izgubilo 70% vbrizganega cisplatina, je v tumorju ostal koncentrirani cisplatin, posledično pa je

bil visok tudi koncentracijski gradient.

V literaturi lahko najdemo podatke, koliko molekul bleomicina potrebujemo, da pride do celične smrti

(Tounekti et al. 1993; Tounekti et al. 2001). Podobno analizo bi bilo potrebno narediti tudi s cisplatinom,

33


kar bi nam omogočilo optimizacijo parametrov zdravljenja za čim višjo znotrajcelično koncentracijo

cisplatina. Z induktivno sklopljeno plazmo z masno spektrometrijo lahko zelo natančno izmerimo maso in

posledično število molekul v celicah. Če obenem pod enakimi pogoji s testom klonogenosti določimo še

preživetje celic, lahko sklepamo na število molekul cisplatina, potrebnih za celično smrt.

Pri elektroporaciji lahko povišanje prevodnosti celične membrane in povišanje njene prepustnosti

razumemo kot dva različna pojava (Miklavčič in Towhidi 2010; Leguèbe et al. 2014). Začetna ideja je bila,

da bi ta pojava poskušali povezati in s spremembo električne prevodnosti celice napovedati prepustnost

membrane, kar pa se je izkazalo za težko dosegljivo. Že pri začetnih poskusih smo namreč ugotovili, da pri

isti prevodnosti zunajcelične tekočine, ki torej povzroči enako vsiljeno transmembransko napetost in enako

gostoto por, dobimo bistveno drugačen vnos barvila v celico (Dermol et al. 2016). To torej nakazuje, da je

elektroporacija bolj kompleksna in da samo nastanek por ne določa transporta v celice. Vseeno bi bilo v

prihodnosti zanimivo ugotoviti, ali je povezava električne prevodnosti ter prepustnosti celične membrane

možna in ali je prepustnost v večji meri določena z biološkimi procesi na celični membrani. Električna

prevodnost celične membrane se namreč relativno hitro zmanjša, medtem ko transport poteka še več minut

po dovedenih pulzih. Pri nanosekundnih pulzih so sicer ugotovili, da še nekaj časa po pulzih električna

prevodnost celične membrane ostaja rahlo povišana (Pakhomov et al. 2007) in da trajanje povišane

prevodnosti korelira s trajanjem transporta preko celične membrane. Druga razlaga podaljšanega transporta

je, da električni pulzi povzročijo oksidacijo membrane, kar poruši urejenost lipidnega dvosloja in poveča

prepustnost celične membrane (Benov et al. 1994; Gabriel in Teissié 1995; Maccarrone et al. 1995;

Pakhomova et al. 2012).

Eden izmed problemov obstoječih modelov elektroporacije je pomanjkanje meritev, npr. nastanka in

širjenja lokalnih transportnih območij v koži, električnih lastnosti posameznih tkiv med elektroporacijo v

odvisnosti od parametrov električnih pulzov, prevodnosti lokalnih transportnih območij, konvekcijskega

transporta v tumorjih, prostorske porazdelitve cisplatina v tumorjih pred elektroporacijo in po njej, vezavo

cisplatina in vivo. Obstoječi modeli so namreč zelo kompleksni in vsebujejo veliko število parametrov,

katerih vrednosti so v večini primerov le ocenjene na velikostni razred. Z optimizacijo parametrov modelov

lahko torej najdemo več lokalnih minimumov, ne pa nujno globalnega, kot smo ugotovili v naši študiji

(Dermol-Černe in Miklavčič 2018). Že pri sami geometriji lokalnih transportnih območij, ki nastanejo pri

elektroporaciji, se vrednosti iz literature razlikujejo. Velikosti lokalnih transportnih območij se razlikujejo

za več velikostnih razredov – od 10 μm do več sto mikrometrov (Pliquett et al. 1996; Pliquett et al. 1998;

Gowrishankar et al. 1999a; Vanbever et al. 1999; Pavšelj in Miklavčič 2008a), primanjkuje pa meritev

časovne dinamike širjenja por.

V prihodnosti bi modele, ki smo jih predstavili v tej disertaciji, lahko vpeljali v načrtovanje medicinskih

posegov elektroporacije. Trenutno parametre zdravljenja optimiziramo tako, da zagotovimo, da je celoten

tumor pokrit z dovolj visokim električnim poljem, okoliško tkivo pa je nepoškodovano. V začetku bi lahko

najprej vpeljali matematične modele prepustnosti celične membrane in celične smrti, nato bi vključili

modele transporta, s katerimi bi izračunali doseženo znotrajcelično koncentracijo kemoterapevtika oziroma

točno število molekul znotraj posameznih celic. Lahko bi tudi zmanjšali krčenje mišic in bolečino, kar bi

34


skrajšalo in poenostavilo zdravljenje. Z vključitvijo meritev električnega polja v tkivu (Kranjc et al. 2014;

Kranjc et al. 2015) pa bi se dalo odziv zdravljenja napovedati še bolj točno.

Matematično modeliranje procesov med elektroporacijo ima vsekakor prihodnost. Menim, da opis procesa z

modelom olajša naše razumevanje. Uporaba modelov nam omogoča, da rezultate interpoliramo, saj

izvajanje nešteto poskusov vseh možnih kombinacij ni smiselno tako s časovnega kot tudi s finančnega

vidika. V okviru doktorske disertacije sem odgovorila na večino vprašanj, ki sem si jih postavila na začetku

dela. Med delom pa so se porodila številna nova vprašanja, na katera je v prihodnosti vredno poiskati

odgovor.

35

Razširjen povzetek v slovenskem jeziku - Izvirni prispevki k znanosti

Izvirni prispevki k znanosti V doktorski disertaciji so naslovljeni trije izvirni prispevki k znanosti.

Matematično modeliranje prepustnosti celične membrane in celične smrti

Z matematičnim modeliranjem lahko opišemo verjetnost za prepustnost ali celično smrti na ravni tkiv.

Naredili smo pregled modelov prepustnosti celične membrane s področja elektroporacije evkariontskih

celic in modelov celične smrti s področja elektroporacije evkariontskih in prokariontskih celic. Obstoječi

modeli so večinoma izbrani na podlagi najboljšega ujemanja z rezultati poskusov in ne izhajajo iz

mehanizmov elektroporacije. Parametre naših izbranih modelov smo določili s pomočjo rezultatov

poskusov in vitro. Enostaven prenos optimiziranih modelov z ravni celic na tkiva ni mogoč. Ker moramo za

vsako vrsto tkiva in vsako skupino električnih parametrov posebej optimizirati matematične modele, je

njihova napovedna moč omejena.

Prehod modeliranja z nivoja ene same celice na nivo tkiv, pri čemer je predstavljen realističen

tridimenzionalen model kože nadgrajen z modelom elektroporacije vseh bistvenih delov in sestavnih

plasti

Elektroporacija celic in tkiv spremeni njihove električne lastnosti. V okviru doktorske disertacije smo

modelirali časovno odvisno spremembo električne prevodnosti posameznih celic v koži – korneocitov,

keratinocitov, papilarnega dermisa. Modelirali smo nastanek lokalnih transportnih območij v roženi plasti in

nastanek por na celicah v spodnjih plasteh kože. Iz električnih lastnosti celic smo izračunali enakovredne

električne lastnosti plasti kože – rožene plasti, epidermisa, papilarnega dermisa in žilne plasti. Naše

izračune smo primerjali s tokovnonapetostnimi meritvami med elektroporacijo kože. Z našo metodo lahko

modeliramo električne lastnosti med elektroporacijo katerih koli celic in tkiv. Edini pogoj je, da poznamo

geometrijo celice ter električne lastnosti znotrajcelične tekočine, celične membrane in zunajceličnega

prostora.

Matematično modeliranje transporta molekul (kemoterapevtika) skozi celično membrano na osnovi

modela dvojne poroznosti

Za učinkovito elektrokemoterapijo je nujen vnos zadostne količine kemoterapevtika v celice. V načrtovanju

posegov ob uporabi kritičnega električnega polja izračunamo, ali bo določeno območje uničeno ali ne, pri

tem pa ne upoštevamo transporta kemoterapevtika v tumorske celice. V okviru tretjega izvirnega prispevka

smo modelirali transport pri elektroporaciji tumorjev. Z modelom dvojne poroznosti smo izračunali in vitro

koeficiente prepustnosti celične membrane v odvisnosti od parametrov električnega polja. Model dvojne

poroznosti z diferencialnimi enačbami opiše difuzijo majhnih molekul topljenca skozi celično membrano in

v tkivu. V numeričnem modelu tumorja smo s koeficientom prepustnosti izračunali transport po tumorju pri

elektrokemoterapiji in model primerjali z rezultati vnosa cisplatina v podkožne tumorje na miših. Model je

dobro opisal rezultate poskusov, če smo vanj vključili še koeficient transformacije, s katerim smo

upoštevali razlike v mobilnosti majhnih molekul in vitro ter in vivo. V prihodnosti bi model lahko uporabili

v načrtovanju posegov elektroporacije, kjer bi tako povišali natančnost napovedi celične smrti.

36

Introduction - Electroporation – mechanisms, and use

Introduction

Electroporation – mechanisms, and use When short high voltage pulses are applied to cells and tissues, plasma membrane becomes transiently

permeable to molecules which otherwise cannot freely pass in or out of the cell (Tsong 1991; Weaver 1993;

Kotnik et al. 2012; Rems and Miklavčič 2016; Rems and Miklavčič 2016). The phenomenon is called

electroporation as pores are presumably formed in the cell membrane (Abidor et al. 1979). After

electroporation, pores can reseal, and cells recover – we consider this reversible electroporation. If the

damage to the cell membrane is too extensive or if the homeostasis is severely altered, cells die - we

consider this irreversible electroporation (Jiang et al. 2015a). Electroporation or pulsed electric field

treatment is used in biotechnology (Haberl et al. 2013a; Haberl et al. 2013b; Mahnič-Kalamiza et al. 2014b;

Kotnik et al. 2015), food-processing (Toepfl et al. 2014; Mahnič-Kalamiza et al. 2014b; Blahovec et al.

2017) and medicine (Yarmush et al. 2014). In medicine, it is used in gene electrotransfer (Daud et al. 2008;

Heller and Heller 2015), DNA vaccination (Vasan et al. 2011; Gothelf and Gehl 2012; Calvet et al. 2014;

Trimble et al. 2015), transdermal drug delivery (Denet et al. 2004; Zorec et al. 2013b), irreversible

electroporation as a soft tissue ablation technique (Davalos et al. 2005; Garcia et al. 2010; Deodhar et al.

2011b; José et al. 2012; Cannon et al. 2013; Scheffer et al. 2014a; Jiang et al. 2015a; Ting et al. 2016; Lyu

et al. 2017) and electrochemotherapy (Mali et al. 2013; Miklavčič et al. 2014; Edhemović et al. 2014;

Campana et al. 2014; Mali et al. 2015; Gasbarrini et al. 2015; Bianchi et al. 2016).

Treatment planning of electrochemotherapy and ablation with irreversible

electroporation When treating tissues with electroporation-based medical treatments, we can use predefined electrode

geometries and pulse parameters (Marty et al. 2006; Mir et al. 2006). When tumors are outside the target

parameters, variable electrode geometry can be used. In this case, electrode geometry and pulse parameters

have to be defined for each treatment separately (Kos et al. 2010; Miklavčič et al. 2010; Pavliha et al. 2012;

Linnert et al. 2012; Edhemović et al. 2014). This is done in the process called treatment planning, when

tumour geometry is extracted from medical images (Pavliha et al. 2013), and position and number of

electrodes and electric pulse parameters are optimized to cover a tumour with a sufficiently high electric

field for reversible or irreversible electroporation (Miklavčič et al. 2006a), depending on the chosen

treatment. Simplifications done in the process and addressed in this thesis are the following.

1) For each tissue type, the critical electric field for permeabilization or cell death used in the

treatment planning was experimentally determined (Šel et al. 2005; Čorović et al. 2012). However, in

reality, the transition from non-permeable to permeable cells and from alive to dead is continuous and

should be described with appropriate continuous function.

2) A possible upgrade of treatment plans is also taking into account excitability of tissues. In

electroporation-based medical treatments, also excitable tissues are treated, either intentionally for example

in irreversible electroporation of brain cancer (Salford et al. 1993; Agerholm-Larsen et al. 2011; Linnert et

al. 2012; Garcia et al. 2012), gene electrotransfer (Hargrave et al. 2013; Hargrave et al. 2014; Bulysheva et

37

Introduction - Dielectric properties of electroporated cells

al. 2016) or ablation of the heart muscle (Lavee et al. 2007; Neven et al. 2014b; Neven et al. 2014a), and

gene electrotransfer of skeletal muscles (Aihara and Miyazaki 1998; Heller and Heller 2015), or

unintentionally when excitable tissues are in the vicinity of the target treated area. Unintentionally, we can

affect the neurovascular bundle (Neal et al. 2014; Ting et al. 2016) when treating prostate cancer, neurons

when treating bone metastases (Tschon et al. 2015; Gasbarrini et al. 2015) or spinal cord when treating

tumors in the spine (Tschon et al. 2015). One of the main drawbacks to the treatment of tissues with pulsed

electric fields is the discomfort and pain associated with repeated electrical stimulation (Miklavčič et al.

2005; Županič et al. 2007; Arena and Davalos 2012; Golberg and Rubinsky 2012), the need to administer

muscle relaxants and anaesthesia (Ball et al. 2010) and synchronization with the electrocardiogram (Mali et

al. 2008; Deodhar et al. 2011a; Mali et al. 2015). The neurons responsible for pain sensation (nociceptors)

can be stimulated by electric pulses (Nene et al. 2006; Jiang and Cooper 2011). An important advance of

the treatment would be to determine a point at which maximum permeability of the membrane could be

achieved while minimizing excitation of the exposed excitable tissues.

3) During the delivery of electric pulses, the electric conductivity of tissues increases, which is in

treatment planning described by a phenomenological sigmoid function (Šel et al. 2005; Ivorra et al. 2009;

Essone Mezeme et al. 2012a; Neal et al. 2012). Another possibility to model change in electric conductivity

is to model change in tissues’ dielectric properties by including the electroporation mechanisms, for

example, pore formation. When electric pulses are delivered to the skin, also local transport region

formation could be included in the model. In this way, instead of an experimentally determined change in

conductivity of tissues, we could model and predict it based on the parameters of the delivered pulses.

4) In the treatment planning of electrochemotherapy, we assume that in tumors electric field above

0.4 kV/cm has to be achieved to permeabilize cells. The treatment plan does not take into account transport

of chemotherapeutics. However, the above-critical electric field does not guarantee transport. In

electrochemotherapy with bleomycin, cell death occurs due to the influx of bleomycin in the cell and

consequent DNA breakage (Tounekti et al. 1993; Tounekti et al. 2001).

Dielectric properties of electroporated cells Electroporation changes dielectric properties of cell membranes of single cells. We can assess this change

with dielectric impedance spectroscopy (Abidor et al. 1993; Schmeer et al. 2004), current-voltage

measurements (Pliquett and Wunderlich 1983; Pavlin et al. 2005; Pavlin and Miklavčič 2008; Suzuki et al.

2011), optical methods (Hibino et al. 1991; Griese et al. 2002) or patch clamp (Pakhomov et al. 2007;

Wegner 2015; Yoon et al. 2016; Napotnik and Miklavčič 2017). Electroporation also increases electric

conductivity of cell monolayers (Ghosh et al. 1993; Müller et al. 2003; Stolwijk et al. 2011; García-Sánchez

et al. 2015). Similar measurements were employed in electroporation of tissues. Dielectric properties of

human, animal and plant tissues can be measured by dielectric spectroscopy (Barnes and Greenebaum

2006; Ivorra and Rubinsky 2007; Grimnes and Martinsen 2008; Dean et al. 2008; Zhuang et al. 2012;

Dymek et al. 2014; Dymek et al. 2014; Trainito et al. 2015; Zhuang and Kolb 2015), current-voltage

measurements during electroporation (Pavlin et al. 2005; Cukjati et al. 2007; Ivorra and Rubinsky 2007;

Pavlin and Miklavčič 2008; Ivorra et al. 2009; Chalermchat et al. 2010; Neal et al. 2012; Becker et al. 2014;

Dymek et al. 2015; Dymek et al. 2015), electric impedance tomography (Davalos et al. 2002; Davalos et al.

38

Introduction - Transport during and after electroporation

2004; Granot and Rubinsky 2007; Meir and Rubinsky 2014), and magnetic resonance electric impedance

tomography (Kranjc et al. 2014; Kranjc et al. 2015; Kranjc et al. 2017). The efficiency of electroporation

can be predicted by the change in electric conductivity of tissues (Cukjati et al. 2007; Ivorra et al. 2009;

Neal et al. 2012). When measuring dielectric properties of tissues, secondary effects can affect

measurements – such as change of cell radii (Serša et al. 2008; Sano et al. 2010; Deodhar et al. 2011b;

Calmels et al. 2012), forming of edema (Ivorra et al. 2009), loss of ions from the cells (Pavlin et al. 2005),

vascular lock (Ivorra and Rubinsky 2007; Serša et al. 2008). Electroporation changes specific conductivity

of tissues mostly in the range of the β dispersion (Pliquett et al. 1995; Ivorra and Rubinsky 2007; Garner et

al. 2007; Oblak et al. 2007; Ivorra et al. 2009; Neal et al. 2012; Zhuang et al. 2012; Salimi et al. 2013)

because pore formation in the cell membrane affects cell membrane polarization due to increase in its

electric conductivity.

Transport during and after electroporation Transport during and after electroporation can occur via three different mechanisms: diffusion,

electrophoresis, and endocytosis. During pulses the transport is electrodiffusive (Li and Lin 2011; Sadik et

al. 2013), and after the pulses, it is diffusive (Pucihar et al. 2008). Endocytosis was observed after

application of longer low-voltage pulses (Rols et al. 1995; Antov et al. 2005).

In vitro, transport due to electroporation is relatively easy to determine using fluorescent dyes and

techniques like fluorescent microscopy, flow cytometry, spectrofluorometric measurements (Napotnik and

Miklavčič 2017). In vivo, the transport due to electroporation is challenging to assess. 57Co-bleomycin

(Belehradek et al. 1994), 111In-bleomycin (Engström et al. 1998), 99mTc-DTPA (Grafström et al. 2006), 51Cr-EDTA (Batiuškaitė et al. 2003), gadolinium (Garcia et al. 2012), lucifer yellow (Kranjc et al. 2015)

and cisplatin (Melvik et al. 1986; Čemažar et al. 1998; Čemažar et al. 1999; Ogihara and Yamaguchi 2000;

Čemažar et al. 2001; Serša et al. 2010; Hudej et al. 2014; Čemažar et al. 2015) were already described in

the literature.

When modeling transport, we have to keep in mind that the conditions in vitro and in vivo are different. In

vitro, at a single cell level, the solute surrounds the cells. Solute can pass cell membrane anywhere as long

as the membrane is permeable. The transport in the extracellular space is not limited. If there is enough of

solute in the extracellular space, the transport stops when the cell membrane reseals. The transport through

the membrane can be directly linked to the cell membrane permeability (Mahnič-Kalamiza et al. 2015). In

vivo, the initial solute concentration varies spatially. All the solute available can enter cells before the

membrane reseals, i.e., the extracellular compartment can be locally depleted. The transport of solute occurs

via diffusion and convection (Canatella et al. 2004). Cells are close together which decreases the possible

area for the uptake and decreases the induced transmembrane voltage due to electric field shielding.

Already in spheroids, it was shown that transport of small molecules was spatially heterogeneous (Canatella

et al. 2004; Gibot et al. 2013). Electric pulses cause vasoconstriction (Serša et al. 2008) which limits the

transport of solute in or out of a tumor. Increased interstitial pressure, heterogeneous perfusion, defective

lymphatic system, binding of drugs to non-target molecules and metabolism additionally affect transport in

a tumor (Baxter and Jain 1989; Baxter and Jain 1990; Baxter and Jain 1991; Jain 1999).

39

Introduction - Models of electroporation

It should be kept in mind that an increase in cell membrane conductivity is not the same as the increase in

its permeability (Wegner et al. 2015). An increase in cell membrane conductivity starts immediately after

pulse application (Pavlin and Miklavčič 2008) while the transport due to increased cell membrane

permeability lasts for minutes or even hours after the application of electric pulses (Saulis et al. 1991;

Golzio et al. 2002). Sufficiently high increase in conductivity indicates that the transport will occur (Cukjati

et al. 2007; Ivorra and Rubinsky 2007; Pavlin and Miklavčič 2008; Ivorra et al. 2009; Neal et al. 2012)

Models of electroporationMathematical modeling offers a concise description of biological phenomena using mathematical and

physical laws; it improves our understanding, our predictions, and decreases the number of needed

experiments. In the clinical setting, modeling can help predict the treatment outcome and optimize the

treatment to achieve the best results. In the field of electroporation, different spatial scales of biological

systems are modeled separately: level of molecules (phospholipids), lipid bilayers, cells, and tissues. If in

the process of modeling we do not achieve good agreement with experimental data, this indicates that we

have not modeled all the relevant processes.

Models at the level of molecules

With molecular dynamics simulations, we can model how different molecules and ions behave when

exposed to an electric field (Tieleman 2006; Böckmann et al. 2008; Delemotte and Tarek 2012; Casciola

and Tarek 2016), but we are restrained to the time scale of 100 to 1000 nanoseconds. Molecular dynamics

studies propose that during electroporation water fingers are formed in the cell membrane. Water fingers

then evolve to hydrophilic pores. Molecular dynamics simulations were compared with experimental data –

in simulations it was predicted that siRNA could cross the cell membrane in a few nanoseconds, which was

also shown experimentally (Breton et al. 2012).

Models at the level of lipid bilayers

In polar solvent like water lipids spontaneously form a lipid bilayer. Models of electroporation of lipids

bilayers are of three main groups (Chen et al. 2006a; Pavlin et al. 2008; Rems and Miklavčič 2014). In the

first group are deterministic electromechanical models - hydrodynamic, elastic, hydroelastic, viscoelastic,

where membranes are treated as elastic bodies, and their behavior is described with laws of electrostatic.

This first group of models does not describe electroporation well. In the second group are models of pore

formation which are stochastic and are derived from energy equations (Abidor et al. 1979). Pore formation

and development during pulse application can be described by the Smoluchowski equation. The

Smoluchowski equation is computationally demanding and was simplified by Neu and Krassowska to its

asymptotic form (Neu and Krassowska 1999; Neu and Neu 2009). In the asymptotic form, we assume that

all pores in the cell membrane are of fixed radii. The asymptotic model can be upgraded to include also

increase in pore radii, but it is computationally demanding (Neu et al. 2003; Smith et al. 2004). In the third

group is a kinetic scheme (Neumann et al. 1998; Miklavčič and Towhidi 2010), where lipid bilayer changes

between permeable and non-permeable state (Böckmann et al. 2008). Transport of small molecules through

a single pore can be modeled with the electrodiffusion equation (Granot and Rubinsky 2008; Li and Lin

2011; Smith and Weaver 2011; Movahed and Li 2012; Movahed and Li 2013). It was determined that

40


continuum models of electroporation of lipid bilayers correspond well to molecular dynamics simulations

(Casciola et al. 2016; Rems et al. 2016).

Models at the level of single cells

Lipid bilayer together with proteins forms cell membrane. Cell membrane can be approximated by a

dielectric surrounded by a conductor. At the cell level, there are several models existing – models of

induced transmembrane voltage, pore formation, transport, permeability, cell death, and change in electric

conductivity of cell. Time and spatial change of induced transmembrane voltage can be analytically

calculated for isolated cells of simple shapes (Kotnik and Miklavčič 2000a; Kotnik and Miklavčič 2000b;

Valič et al. 2003) but requires a numerical calculation for other more complicated shapes (Pucihar et al.

2006). When the induced transmembrane voltage is known, pore formation can be described either with

asymptotic equation (Neu and Krassowska 1999; Neu and Neu 2009) or by also including the change in

pore radii (Smith et al. 2004; Smith et al. 2014). Transport of molecules into a single cell can be described

with the Neumann’s kinetic scheme (Neumann et al. 1998; Miklavčič and Towhidi 2010), the

pharmacokinetic model (Puc et al. 2003) or with the electrodiffusion equation (Pucihar et al. 2008; Li and

Lin 2011; Li et al. 2013; Sadik et al. 2013; Sadik et al. 2014; Blumrosen et al. 2016). Electrical properties

of cells can be calculated with an equivalent circuit (Pethig 1984; Gowrishankar and Weaver 2003). Cell

permeability and cell survival can be modeled with mathematical models (Schoenbach et al. 2009; Dermol

2014; Dermol and Miklavčič 2014; Dermol and Miklavčič 2015; Dermol and Miklavčič 2016) which

accurately describe the shape of the permeability and survival curves but are phenomenological and are not

based on any mechanisms of electroporation. Cell permeability and conductivity can be modeled separately

as two phenomena (Miklavčič and Towhidi 2010; Leguèbe et al. 2014).

Models at the level of cell clusters and tissues

Several cells together form either a monolayer or a tissue. Current tissue models are of two types – they

either model the tissue as a bulk (Šel et al. 2005; Čorović et al. 2007; Pavšelj et al. 2007; Pavšelj and

Miklavčič 2008b; Golberg and Rubinsky 2010; Neal et al. 2012; Čorović et al. 2012; Garcia et al. 2014) or

also model the tissues' microscopic structure (Gowrishankar and Weaver 2003; Stewart Jr. et al. 2005;

Gowrishankar and Weaver 2006; Esser et al. 2007; Joshi et al. 2008; Esser et al. 2009; Essone Mezeme et

al. 2012b; Dymek et al. 2015).

The bulk tissue models describe the electric field distribution as a function of tissue properties, applied

voltage and the geometry of the electrodes. Transport to cell clusters or tissues was modelled by statistical

models which describe the data well but do not include specific transport mechanisms (Canatella and

Prausnitz 2001), the kinetic scheme of electroporation (Neumann et al. 1998; Miklavčič and Towhidi 2010),

the pharmacokinetic model (Puc et al. 2003; Agarwal et al. 2009), the electrodiffusion equation or the dual-

porosity model (Mahnič-Kalamiza et al. 2014a; Mahnič-Kalamiza et al. 2015). The dual porosity model

(Mahnič-Kalamiza et al. 2014a; Mahnič-Kalamiza et al. 2015) is based on the diffusion equation, connects

pore formation with membrane permeability, and includes transport between several compartments - in our

case between tumor cells, interstitial fraction and peritumoral environment and has the possibility to include

also thermal relations (Mahnič-Kalamiza et al. 2017). Boyd and Becker also used the dual porosity model

41


by combining it with equations for the degree of electroporation, change of electrical conductivity, and

membrane resealing (Boyd and Becker 2015; Argus et al. 2017).

The models in the second group include properties of single cells and thus local inhomogeneities. If tissues

are made of spheroidal cells in low-volume fractions, their effective dielectric properties can be calculated

using the Maxwell-Wagner, the Hanai-Bruggeman, and other approximations (Asami 2002). Tissue was

sampled using Cartesian transport lattices and current, voltage, and electric charge through the lattice were

calculated using the Kirchhoff’s laws (Gowrishankar and Weaver 2003; Stewart Jr. et al. 2005;

Gowrishankar and Weaver 2006; Esser et al. 2007; Esser et al. 2009). The equivalent circuit was used to

analyze the effects of electroporation on cells and then averaged to obtain results for tissues (Ramos et al.

2003; Ramos 2005). Tissue was modeled with the two-dimensional Voronoi network, and the induced

transmembrane voltage was calculated in each region (i.e., cell) (Joshi et al. 2008) The weakness of these

models is, however, that they are in two-dimensions. Besides, in tissues, cells are packed close together, the

local electric field decreases due to electric field shielding, and consequently, the induced transmembrane

voltage is lower than in isolated cells (Susil et al. 1998), there are the blood flow (Serša et al. 2008), and the

immune system (Serša et al. 1997; Neal et al. 2013; Serša et al. 2015) present. In 3-dimensions, tissue was

numerically modeled as an infinite lattice of spherical cells (Pavlin et al. 2002; Ramos 2010). The cells

were all of the same size, electrically unconnected and in an iterative arrangement which is not in

agreement with the circumstances in tissues. Besides, maximal volume fraction was lower than the ones we

encounter in most tissues. The model was upgraded using randomly distributed spheres of different sizes to

model change in electric conductivity of the cell membrane and the fraction of permeabilized cells (Essone

Mezeme et al. 2012b). A similar model was built for the description of electroporation of spinach leaves.

The authors adapted the shapes and sizes of cells to the morphology of a spinach leaf; however, they

modeled only a small part of a leaf (Dymek et al. 2015). The model by Huclova et al. (Huclova et al. 2010;

Huclova et al. 2011; Huclova et al. 2012) offers a more realistic description since cells can be of different

shapes, sizes, and distances between them can be adapted. The dielectric properties of cells are then

averaged to obtain properties on the tissue level. The model by Huclova et al. describes the behavior of cells

in the linear range, but we have added electroporation to it (Dermol-Černe and Miklavčič 2018). Instead of

numerical calculations, the properties of tissues can be calculated analytically. The frequency dependent

dielectric properties of tissues can be described using phenomenological models – the Debye, the Cole-

Cole, and other models (Gabriel et al. 1996a; Gabriel et al. 1996b; Gabriel et al. 1996c; Asami 2002).

42

Aim

Aim The aim of this thesis can be summarized in two points:

1) Use mathematical models to model cell membrane permeability and depolarization and cell death;

2) Include mechanisms of electroporation in models of change in dielectric properties of tissues and

transport due to electroporation.

In the scope of the first aim, we fitted several mathematical models to experimental data of cell membrane

permeability and cell death. We predicted the percentage of permeable cells, exposed to the inhomogeneous

electric field, from the experiments on cells, exposed to the homogeneous electric field. We used optimized

mathematical models of cell membrane permeability. We compared permeability and depolarization of cell

membranes of excitable and non-excitable cells and modeled both phenomena. We made a review of the

existing cell death models, usually employed in prediction of liquid food pasteurization and sterilization,

and fitted the existing models to our experimental data of cell death caused by irreversible electroporation.

We focused on time dynamics of dye uptake due to increased cell membrane permeability and the

correlation between the membrane resealing and the transport of dye to cells.

In the scope of the second aim, we focused on two important phenomena present during electroporation –

change in dielectric properties of cells and tissues and increased transport of molecules in and out of the

cells. We studied the change in dielectric properties of skin when either long low-voltage or short high-

voltage pulses were applied. We determined the contribution of local transport regions in the stratum

corneum and the pore formation in the lower layers to the overall increase in electric conductivity. We

modeled the pore formation with the asymptotic pore model and correlated the pore formation and local

transport region formation to the shape of the current-voltage measurements in the Franz diffusion cell.

We also studied the transport of small molecules (e.g., cisplatin) in the tumor tissue after electroporation.

We modeled cisplatin uptake by tumor cells in mouse melanoma tumors. Animal experiments take a long

time, a limited number of parameters can be tested due to ethical issues, and they are expensive. Thus, we

aimed to determine if experiments on cell level can be used to predict the treatment outcome in animal

experiments with a possibility of later translation to medical treatments. We investigated the contribution of

close cell contacts, high cell volume fraction and smaller available area of cell membrane for transport in

vivo. We determined that cisplatin uptake in vivo can be determined from the cisplatin uptake in vitro,

provided we take into account reduced mobility of the molecules.

43

Results and Discussion

Results and Discussion In the present thesis, the aim was to link experiments on cell level (in vitro) to experiments on tissue level

(in vivo) by using models describing electroporation-related processes at a single cell level and processes at

the tissue level. Three original scientific contributions were made:

1. Mathematical modeling of cell membrane permeability and cell death;

2. The transition between single-cell models and the tissue models, where the realistic three-

dimensional skin model is presented with the model of electroporation of all components and

essential layers of the skin.

3. Mathematical modeling of transport of small molecules (chemotherapeutic) across the cell

membrane using the dual porosity model.

The work presented in this thesis consists of two parts – experimental in a cell culture laboratory and

modeling. I believe that only by using modeling we can understand experimental results and vice versa –

models are only relevant when they are based on experimental measurements. Models based on a good

experimental data facilitate our understanding of the modeled phenomena, enable us to make predictions

about the behavior of biological systems and decrease number of needed experiments.

The work done in the scope of this thesis was presented in six papers and a book chapter. There, the results

are presented and thoroughly discussed. Thus, the Results and Discussion section consist of the published

papers, and the thesis will continue with a Conclusion and Future Work section. A summary of the work is

the following.

The first contribution was addressed in four papers and one book chapter.

Paper 1 (Dermol and Miklavčič 2014) entitled Predicting electroporation of cells in an inhomogeneous

electric field based on mathematical modeling and experimental CHO-cell permeabilization to

propidium iodide determination described mathematical models of cell membrane permeability. We

focused on four models – the symmetric and asymmetric sigmoid, the hyperbolic tangent and the Gompertz

curve. From experimental data of the cell membrane permeability obtained on attached cell monolayers

exposed to the homogeneous electric field, we predicted cell membrane permeability in the inhomogeneous

electric field. Prediction of permeability was possible if the density of the cells in the homogeneous field

was comparable to the density of cells in the inhomogeneous field. We conclude that models describing cell

electroporation probability could be useful for development and presentation of treatment planning for

electrochemotherapy and non-thermal irreversible electroporation.

Paper 2 (Dermol and Miklavčič 2015) entitled Mathematical Models Describing Chinese Hamster

Ovary Cell Death Due to Electroporation In Vitro focused on mathematical models of cell survival. We

exposed cell suspension to pulses of parameters similar to irreversible electroporation parameters - several

tens of 100 μs long pulses of up to 4 kV/cm, delivered at the repetition frequency 1 Hz. Cell survival was

evaluated with the clonogenic assay. Several models of survival previously used in the literature were fitted

to our experimental results. The models were: the first-order kinetics, the Hülsheger, the Peleg-Fermi, the

44


Weibull, the Logistic, the adapted Gompertz, and the Geeraerd model. The most appropriate models of cell

survival as a function of the treatment time were the adapted Gompertz and the Geeraerd models and, as a

function of the electric field, the logistic, the adapted Gompertz, and the Peleg-Fermi models. The next

steps to be performed are a validation of the most appropriate models on tissues and determination of the

models’ predictive power. A review of cell death models of the eukaryotic and prokaryotic cells is written

in the book chapter (Dermol and Miklavčič 2016) entitled Mathematical Models Describing Cell Death

Due to Electroporation.

Paper 3 (Dermol-Černe et al. 2018) entitled Plasma membrane depolarization and permeabilization due

to electric pulses in cell lines of different excitability consists of the experimental part and modeling part

regarding the cell depolarization and cell membrane permeability. The aspect of excitability is important

when treating excitable tissues with electroporation or when excitable tissues are in the vicinity of the

treated area. The experimental strength-duration curve to one pulse of length 10 ns – 10 ms was determined

for excitable and non-excitable cells and then modeled with the Lapicque curve and the Hodgkin-Huxley

model. The Hodgkin-Huxley model gave insight into the ability of the short bipolar to permeabilize but not

excite the tissue. The permeability curve to one pulse of length 100 ns – 100 ms was obtained from the

literature along with the Saulis model, and was additionally modeled with the Schwann equation. The

depolarization threshold was higher for the excitable cells than for the non-excitable cells. All four cell lines

responded similarly to pulses of standard electrochemotherapy parameters, and their permeability curves

were modeled with the symmetric sigmoid. Thus, electroporation is a feasible means of treating excitable

and non-excitable cells with pulses of similar parameters. Our results show the potential of achieving cell

membrane permeability and minimizing or avoiding excitation/pain sensation which needs to be explored in

more detail. In future studies, it should be established, to what extent the in vitro depolarization and

excitability correlate to the actual excitation and pain sensation in vivo.

Paper 4 (Dermol et al. 2016) entitled Cell Electrosensitization Exists Only in Certain Electroporation

Buffers consists of experimental work on the effect of pulse delivery in one or several trains on the uptake.

Electroporation-induced cell sensitization was described as the occurrence of delayed hypersensitivity to

electric pulses caused by pretreating cells with electric pulses. It was achieved by increasing the duration of

the electroporation treatment at the same cumulative energy input. It could be exploited in electroporation-

based treatments such as electrochemotherapy and tissue ablation with irreversible electroporation. The

mechanisms responsible for cell sensitization, however, have not yet been identified. We investigated cell

sensitization dynamics in five different electroporation buffers. We split a pulse train into two trains varying

the delay between them and measured the time dynamics of propidium uptake by fluorescence microscopy.

By fitting the first-order model to the experimental results, we determined the uptake due to each train (i.e.,

the first and the second) and the corresponding resealing constant. Cell sensitization was observed in the

growth medium but not in other tested buffers. The effect of pulse repetition frequency, cell size change,

cytoskeleton disruption and calcium influx do not adequately explain cell sensitization. Based on our

results, we can conclude that cell sensitization is a sum of several processes and is buffer dependent.

Further research is needed to determine its generality and to identify underlying mechanisms.

The second contribution was addressed in one paper.

45


Paper 5 (Dermol-Černe and Miklavčič 2018) entitled From cell to tissue properties – modeling skin

electroporation with pore and local transport region formation modeled skin electroporation by pore

formation and local transport region formation at the level of single cells and skin layers. We showed that it

is possible to model electroporation at the level of single cell and then generalize the results to the bulk

tissue where calculations are considerably faster. Current models of tissue electroporation either describe

tissue with its bulk properties or include cell level properties, but model only a few cells of simple shapes in

low-volume fractions or are in two dimensions. We constructed a 3-dimensional model of realistically

shaped cells in realistic volume fractions. By using a ‘unit cell’ model, the equivalent dielectric properties

of whole tissue could be calculated. We calculated the dielectric properties of electroporated skin. We

modeled electroporation of single cells by pore formation on keratinocytes and on the papillary dermis

which gave dielectric properties of the electroporated epidermis and papillary dermis. During skin

electroporation, local transport regions were formed in the stratum corneum. We modeled local transport

regions and increase in their radii or density which affected the dielectric properties of the stratum corneum.

The final model of skin electroporation accurately described measured electric current and voltage drop on

the skin during electroporation with long low-voltage pulses. The model also accurately described voltage

drop on the skin during electroporation with short high-voltage pulses. However, our results indicated that

during application of short high-voltage pulses additional processes might occur which increased the

electric current. Our model connects the processes occurring at the level of cell membranes (pore

formation), at the level of a skin layer (formation of local transport region in the stratum corneum) with the

tissue (skin layers) and even level of organs (skin). Using a similar approach, electroporation of any tissue

can be modeled, if the morphology of the tissue is known.

The third contribution was addressed in one paper.

Paper 6 (Dermol-Černe et al. 2018, submitted) entitled Connecting the in vitro and in vivo experiments

in electrochemotherapy: Modeling cisplatin transport in mouse melanoma by the dual-porosity

model was a modeling study where transport of chemotherapeutic in the tumor cells was modeled by the

dual-porosity model based on experimental results obtained on cell suspension. In electrochemotherapy two

conditions have to be met to be successful – the electric field of sufficient amplitude and sufficient uptake

of chemotherapeutics. Current treatment plans only take into account the critical electric field to achieve

cell membrane permeability. However, increased permeability of cell membrane alone does not guarantee

uptake of chemotherapeutics and consequently successful treatment. In our study, we described the

transport of cisplatin in vivo based on experiments performed in vitro. In vitro, a spectrum of parameters

can be explored without ethical issues. In the experimental part of our study, we performed in vitro and in

vivo experiments. Mouse melanoma B16-F1 cell suspension and inoculated B16-F10 tumors were exposed

to electric pulses in the presence of the chemotherapeutic cisplatin. The uptake of cisplatin was measured

by the inductively coupled plasma mass spectrometry. In modeling part of our study, we modeled the

transport with the dual-porosity model which is based on the diffusion equation, connects pore formation

with membrane permeability, and includes transport between several compartments. In our case, there were

three compartments - tumor cells, interstitial fraction and peritumoral region. Our hypothesis was that in

vitro permeability coefficient can be introduced in vivo, as long as tumor physiology is taken into account.

46


A transformation from in vitro to in vivo was possible by introducing a coefficient of transformation which

takes into account in vivo characteristics, i.e., the smaller available area of the plasma membrane for

transport due to high cell density, and presence of cell-matrix in vivo, thus reducing drug mobility. Our

model offers a step forward to linking transport models at the cell level (in vitro) to the tissue level (in

vivo).

47


48

Results and Discussion - Paper 1

Paper 1 Title: Predicting electroporation of cells in an inhomogeneous electric field based on mathematical

modelling and experimental CHO-cell permeabilization to propidium iodide determination

Authors: Janja Dermol and Damijan Miklavčič

Publication: Bioelectrochemistry, vol. 100, pp. 52–61, Dec. 2014.

DOI: 10.1016/j.bioelechem.2014.03.011

Impact factor: 4.172 (2014), 3.854 (5-year)

Quartile: Q1 (Biology, Biophysics, Electrochemistry)

Rank: 13/85 (Biology), 15/73 (Biophysics), 5/28 (Electrochemistry)

49

Bioelectrochemistry 100 (2014) 52–61

Contents lists available at ScienceDirect

Bioelectrochemistry

j ourna l homepage: www.e lsev ie r .com/ locate /b ioe lechem


Predicting electroporation of cells in an inhomogeneous electric fieldbased on mathematical modeling and experimental CHO-cellpermeabilization to propidium iodide determination

Janja Dermol, Damijan Miklavčič ⁎University of Ljubljana, Faculty of Electrical Engineering, Tržaška 25, SI-1000, Ljubljana, Slovenia

⁎ Corresponding author. Tel.: +386 1 4768 456.E-mail address: [email protected] (D. Mik

http://dx.doi.org/10.1016/j.bioelechem.2014.03.0111567-5394/© 2014 Elsevier B.V. All rights reserved.

50

a b s t r a c t
a r t i c l e i n f o
Article history:Received 17 July 2013Received in revised form 13 March 2014Accepted 24 March 2014Available online 29 March 2014

Keywords:ElectroporationMathematical modelingPropidium iodideCell densityCHO cellsElectric field distribution

High voltage electric pulses cause electroporation of the cell membrane. Consequently, flow of the moleculesacross the membrane increases. In our study we investigated possibility to predict the percentage of theelectroporated cells in an inhomogeneous electric field on the basis of the experimental results obtained whencells were exposed to a homogeneous electric field. We compared and evaluated different mathematical modelspreviously suggested by other authors for interpolation of the results (symmetric sigmoid, asymmetric sigmoid,hyperbolic tangent and Gompertz curve). We investigated the density of the cells and observed that it has themost significant effect on the electroporation of the cells while all four of the mathematical models yieldedsimilar results. We were able to predict electroporation of cells exposed to an inhomogeneous electric fieldbased on mathematical modeling and using mathematical formulations of electroporation probability obtainedexperimentally using exposure to the homogeneous field of the same density of cells. Models describing cellelectroporation probability can be useful for development and presentation of treatment planning forelectrochemotherapy and non-thermal irreversible electroporation.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

High voltage electric pulses affect membrane's selective permeabili-ty. According to the theory electroporation of the membrane occurs aspores are formed in the membrane. Cell membrane thus becomes per-meable for different molecules which otherwise cannot pass throughthe membrane in or out of the cell [1–3]. Increased membrane perme-ability occurs in the regions of the membrane where the transmem-brane voltage exceeds a certain threshold, which is characteristic ofeach cell line, but also depends on pulse parameters [4,5]. If the cell isable to recover after the exposure to electric pulses we call this revers-ible electroporation. If the cell cannot recover and it does not surviveelectroporation we call it irreversible [6]. Electroporation is widelyused in different areas—gene transfer [7–9], cancer treatment [10–12],biotechnology [13,14] and food processing [15–17].

When predicting the electroporation of cells, for example in a tissue,it is usually (implicitly) assumed that cells are electroporated if the in-duced transmembrane voltage exceeds the characteristic threshold[18,19]. If the induced transmembrane voltage is below the characteris-tic threshold the cells are not electroporated. In reality the transitionfrom non-electroporated to reversibly and irreversibly electroporated

lavčič).

state is continuous. So far differentmathematicalmodels of electropora-tion have been proposed in the literature [20–22]. With the appropri-ately chosen mathematical model we could predict percentage of cellsaffected if a certain voltage is applied using specific electrode geometry.

In recent years electroporation based treatments have paved theirway to clinical use. Electrochemotherapy for solid cutaneous, subcuta-neous tumors and metastasis [11,23], as well as for deep seated tumors[24,25] is being used in clinics. Minimally invasive non-thermal irre-versible electroporation as soft tissue ablation has also been proposed[6] and used in animal [26] and human clinics [27]. In all these casestreating deep seated tumors or soft tissue using minimally invasiveprocedures a need for pretreatment planning was clearly established[28–30]. Until now visualization of electric field distribution is usedas being the most important predictor of tissue permeabilization andablation [25,29,31,32].

There is a considerable number of studies of electroporation in densecell suspensions available [33–36], but there aremuch less studies avail-able on tissues [37–39]. In these latter studies besides the density of thecells, applied electric field, cell line, and cells' mutual electric shielding,the connections between the cells and their irregular shape could playan important role as well [4]. In our present study the experimentswere performed on monolayers of cells of different densities.

When defining the duration and the voltage of the applied pulses wemodel the geometry of the electrodes and of the cells as a bulk 2D “tissue”layer. Electric field (E field) distribution is numerically calculated and

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53J. Dermol, D. Miklavčič / Bioelectrochemistry 100 (2014) 52–61


adequate voltage is determined [40]. Because of the complexity of the tis-sues and electrodes analytical calculations are usually in most cases notpossible. There is no conventional or easy way to measure E field in vivo.We have recently proposed a method based on current density imaging(CDI) and magnetic resonance electrical impedance tomography(MREIT) to measure E field in biological tissues [41,42]. However, themost reliable or eveneasyway is numericalmodelingwhichweemployed.

Until now, the calculated electric field is thresholded with two dif-ferent thresholds to obtain three areas—the area where irreversibleelectroporation occurs, the areawhere reversible electroporation occursand the area where electroporation does not occur. The cell responsecan be approximated with a step function. Electroporation is 100% ifthe applied electric field is above the characteristic electroporationthreshold and 0% if it is below. In the same way area where irreversibleelectroporation occurs can be modeled. Irreversible electroporation is100% if the applied electric field is above the characteristic thresholdfor irreversible electroporation and 0% if it is below.

The aim of our study was to predict a percentage of electroporatedcells grown as a dense monolayer exposed to an inhomogeneous field.We performed experiments in a homogeneous electric field and deter-mined the percentage of cell electroporation for a certain applied E.We used these results in four different mathematical models, whichinterpolated and extrapolated the percentage of electroporated cells toother values of E. We used homogeneous E for calculating parametersof the mathematical models because it is possible to determine thepercentage of electroporated cells at a certain E. We validated themathematical models by exposing cells to the inhomogeneous fieldand comparing predicted and experimentally determined values of per-centage of electroporation. We used the inhomogeneous electric fieldfor validation because in tumors and tissues the electric field aroundthe electrodes is in almost all cases inhomogeneous. The predictedvalues were obtained by using the numerically calculated inhomoge-neous E in mathematical models. We obtained the percentage of elec-troporation in the dependence on E for the area around the electrodes.

The advantage of this approach is that simple electrode geometryconfigurations can be used to calculate the parameters of the mathe-matical models. Mathematical models predicting cell electroporationcan then be applied to arbitrary electrode geometry. This kind of math-ematical relationship could allow us to present treatment plans in aclearer and more understandable way. At the moment, the treatmentplans present the E field applied to a certain area of a tissue/tumor.Using this method the percentage of the electroporated area of a tis-sue/tumor could be shown. Eventually also the number and durationof pulses could be taken into account [43].

The block scheme on Fig. 1 shows the procedure used in our study.First, we exposed cells to the homogeneous field (block 1). Based onthese results we determined the parameters of a mathematical modelof cell electroporation (block 2). We used this mathematical modeland inhomogeneous E field distribution to predict the percentage ofelectroporation (block 3). We then validated the model by exposingcells to the inhomogeneous field (block 4) by comparing predicted

electric field

cell exposed tohomogeneous E

E field

distributi

mathematicamodel of celelectroporati

1 2

Fig. 1. A block scheme showing the layout of the article. Numbe

and experimental values. If the results were not in good agreement,we adjusted themathematicalmodel (block 2) and repeated the predic-tion of electroporation and the validation of the mathematical model.

2. Methods

2.1. Cell line and cell culture

A series of experiments were performed on Chinese hamster ovarycells (CHO). Cells were grown in monolayers of different densities inPetri dishes 40 mm in diameter (TPP, Switzerland) in 2 mL of culturemedium (Ham's Nutrient Mixtures HAM-F12, Sigma-Aldrich, Steinheim,Germany) supplemented with 10% FBS, antibiotics and L-glutamine for2 days at 37 °C and 5% CO2 (Kambič, Slovenia).

2.2. Pulse generator and pulse exposure

For the experiments electric pulse generator Cliniporator (IGEA,Italy) was used. For each experiment one rectangular pulse in the dura-tion of 1mswas applied. The voltages were chosen so that themaximalelectric field in the homogeneous field was 1.6 kV/cm and in the inho-mogeneous field it was 2.3 kV/cm. Therefore, for the homogeneousfield the applied voltages were between 200 V and 800 V with a stepof 200 V while for the inhomogeneous field the applied voltages werebetween 80 V and 140 V with a step of 20 V. Two different electrodeconfigurations were used. For the homogeneous field two parallel Pt/Irwire electrodes with 0.75 mm diameter and distance between theinner edges of the electrodes set at 5 mm [44] were positioned at thebottom of the Petri dish (Fig. 2a). For the inhomogeneous field weused one needle electrode pair with diameter 0.5 mm and distance be-tween the inner edges of the electrodes set at 1.0 mm (Fig. 2b) [45].

Cells were exposed to the electric pulses in a medium of 1 mL HAM-F12 and 100 μL of 1.5 mM propidium iodide (PI). PI is a non-permeantfluorescent dye, which emits strong fluorescence after entering the celland thus allows easy determination of cell electroporation and discrim-ination between electroporated and non-electroporated cells bythresholding fluorescent images (see 2.3 Fluorescence microscopy).After the exposure to an electric pulse the cells were incubated for 5min at room temperature and then washed with HAM-F12.

2.3. Fluorescence microscopy

The cells were observed by an inverted microscope AxioVert 200(Zeiss, Germany) under 10× magnification. In each experiment in thehomogeneous field up to five phase contrast and five correspondingfluorescent images were acquired from randomly selected fields ofview between the electrodes. In the experiments in the inhomogeneousfield four imageswere acquiredwhen 80Vwas applied and nine imageswhen more than 80 V was applied to obtain the area around the elec-trodes. The number of experiments and images acquired in each exper-iment for each E for homogeneous field is shown in Table 1. The number

percentage of permeabilization

cell exposed toinhomogeneous E

on

l l on as f(E)

modelvalidation

4

3

rs in the blocks show the temporal sequence of our study.

51

Fig. 2. Photos of electrodes, used in the experiments. a—Two parallel Pt/Ir wire electrodeswith 0.75 mm diameter and distance between their inner edges set at 5 mm. b—One nee-dle Pt/Ir electrode pairwith diameter 0.5mm and distance between the inner edges of theelectrodes set at 1.0 mm.

Table 2Number of experiments (one petri dish and one composed image for oneexperiment) in the inhomogeneous field.

Voltage applied (V) Number of experiments

80 9100 12120 6140 8

54 J. Dermol, D. Miklavčič / Bioelectrochemistry 100 (2014) 52–61


of images used at different cell densities is also reported in Table 1. Forthe inhomogeneous field the number of experiments is reported inTable 2. In each experiment one pair of composed images was acquired(one fluorescent and one phase-contrast image). Fluorescent andphase-contrast images were then stacked together in Adobe PhotoshopCS5 (Adobe Systems Inc., San Jose, CA, USA) to get one image of thewhole area around the electrodes (Fig. 3a,b). For acquisition we usedthe VisiCam 1280 camera (Visitron, Germany) and MetaMorph PC soft-ware (Molecular Devices, USA).

2.4. Fitting of the mathematical models

First, the acquired fluorescent images from experiments in thehomogeneous electric field were thresholded and electroporated cellswere manually counted. Cells in the phase contrast images were manu-ally counted as well. The percentage of the electroporated cells andstandard deviation were calculated. The results were classified intothree groups on the basis of the density of the cells i.e. on the numberof the cells on the phase contrast images. The first groupwas the imageswith less than 300 cells, second was the images with density in therange from 300 to 600 cells and the third was the images with morethan 600 cells per image. In Fig. 4 we can see how different groups ofcell densities look like under the microscope. Fig. 4a shows a typicalexample of a least dense monolayer (less than 300 cells per image),Fig. 4b shows the typical example of a medium dense cell monolayer(from 300 to 600 cells per image) and Fig. 4c shows the densest mono-layer (more than 600 cells per image).

Differentmathematicalmodels (symmetric sigmoid (Eq. (1)), asym-metric sigmoid (Eq. (2)), Gompertz curve (Eq. (3)) and hyperbolic tan-gent (Eq. (4))) suggested previously by other authors [20,46,47] werefitted in Matlab (R2010a, Mathworks, USA) to the experimental dataof the percentage of electroporated cells in the homogeneous fieldusing themethod of non-linear least squares. We obtained an analytical

Table 1Number of experiments for each voltage in the applied homogeneousfield. In each experiment apairs (phase-contrast and fluorescent) for each density of the cells is reported in the 3rd, 4th a

Applied voltage (V) Number of experiments Number of images (N600 cells/image)

200 6 14300 4 4400 8 11500 9 6600 6 10700 6 9800 8 7

52

expression with optimized parameters for each of the equations(Eqs. (1)–(4).

p Eð Þ ¼ 1= 1þ exp − E−E50%ð Þ=bÞð Þ�100%� ð1Þ

p Eð Þ ¼ 1þ v� exp − E−E50%ð Þ=bð Þ� �∧ −1=vð Þ�100% ð2Þ

p Eð Þ ¼ exp − exp − E−E50%ð Þ=bð Þð Þ�100% ð3Þ

p Eð Þ ¼ 1þ tanh B� E−E50%ð Þ� �� =2�100% ð4Þ

E50% represents the electric field at which 50% of the cells areelectroporated and p denotes the percentage of electroporated cells inthe dependence of the applied electric field. Parameters b and B definethe width of the curve, e.g. how quickly the cells go from the non-electroporated state to the electroporated state when the electric fieldis increasing. Parameter v defines the slope of the Eq. (2). E in all fourequations means the applied electric field.

2.5. Modeling of the inhomogeneous electric field

Numerical calculations of the distribution of the inhomogeneouselectric field were performed in COMSOL Multiphysics (version 3.5,COMSOL, Sweden). Themodel wasmade in the AC/DCmodule. Theme-dium around the electrodes had its electrical conductivity set at 1.0 S/mwhich is the conductivity of the pulsing buffer. Since the pulse duration(1 ms) was much longer than a typical constant for polarization of thecell membrane (around 1 μs) [48] steady-state analysis was made. Thecalculated distribution of the electric field around the electrodes isshown in Fig. 5a.

The area around the electrodes in the inhomogeneous electric fieldwas divided using contours into subareas where the electric field waswithin a certain range. An example of these contours is presented inFig. 5b. When 80 V was applied the area was divided into 4 subareas,when 100 V was applied the area was divided into 5 subareas andwhen 120 V or 140 V was applied the area was divided into 6 subareas.The minimal electric field still analyzed was 0.30–0.39 kV/cm. Thelimits of the analyzed ranges of electric field for all of the applied volt-ages are presented in Table 3. Table 3 shows also the percentage of

t least three imageswere acquired from randomly chosen points of view. Number of imagend 5th columns.

Number of images (300–600 cells/image) Number of images (b600 cells/image)

10 56 38 16

25 208 107 6

22 7

Fig. 3. a—Phase contrast composed image from under the microscope, 10×magnification,approximate position of the electrodes is marked with black circles, polarity of the elec-trodes ismarkedwith+ and− signs. b—Composed fluorescent image from under themi-croscope, 10×magnification, averaged 6 images, applied one pulse of 120 V, approximateposition of the electrodes ismarkedwithwhite circles, polarity of the electrodes ismarkedwith + and− signs.



the whole analyzed area without the electrodes taken by each ofthe ranges of electric field when different voltages are applied. Up to0.8 kV/cm the limits are the same for all the applied voltages. The closerwe get to the electrodes, the faster the electric field is increasing. Limitsof electric field values in the inhomogeneous field were based on the

Fig. 4. Phase contrastmicroscopic imageswith different numbers of cells. a—Image belongs to thto 600 cells per image. c—Image belongs to the density group of more than 600 cells per image

size of the area between two contours. The upper and lower limits ofthe area were set so that the areas between two contours were approx-imately the same in size. Therefore, the limits of the areas are not thesame for different voltages applied (Table 3). For each of the appliedvoltages the corresponding contours (limits of the ranges of electricfield) were superimposed to the microscopic images. An example ofcontours superimposed to the fluorescent image can be seen in Fig. 5c.The maximal values from Table 3 are based on a numerical calculationwhere this was the highest electric field value achieved when corre-sponding voltage was applied.

For each subarea from Table 3 cells on phase contrast images and onthresholded fluorescent images (Fig. 6a) were manually counted. Thepercentage of the electroporation and standard deviation in each areawere determined.

In Comsol we transformed the calculated inhomogeneous electricfield into the predicted percentage of the electroporated cells in thedependence on the geometry (Fig. 6b). We achieved transformationby using the numerically calculated inhomogeneous electric field values(Fig. 5a) in mathematical models (Eqs. (1)–(4)) with optimized coeffi-cients to determine the predicted percentage of electroporated cells.

From the continuous distribution of the expected percentage of theelectroporated cells (Fig. 6b) we determined areas with ranges of pre-dicted percentage of electroporated cells. These areas had the samesize and shape as the subareas of electric field around the electrodes(Fig. 5b). The values of the borders of these subareas were determinedempirically. We put the image of areas with ranges of predicted per-centage of electroporation on top of the image with areas with rangesof the electric field. We determined at which values of the borders theoverlapping was complete. This allowed us to compare the predictedpercentage of electroporation and the experimentally determined per-centage of electroporation in the same subarea. The comparison wasmade for each of the mathematical models (Eqs. (1)–(4)) with opti-mized coefficients.

3. Results

We first determined the percentage of the electroporated cells ex-posed to a homogeneous electric field, and determined the influenceof the cell density by fitting different mathematical models to the exper-imental data. Then we used the model with the best fit (highest R2) topredict cell electroporation in the inhomogeneous field. Predicted valueswere compared to experimentally determined cell permeabilization inthe inhomogeneous field.

3.1. Cell electroporation in a homogeneous electric field

Each of the four proposed mathematical models of electroporation(Eqs. (1)–(4)) was fitted to the experimental data from all three celldensity groups, i.e. less than 300 cells per image, 300 to 600 cells per

e density group of under 300 cells per image. b—Image belongs to the density group of 300.

53

image of Fig.�3

image of Fig.�4

E(appl) / (kV/cm)

a

b

c



54

image and more than 600 cells per image. R2 value and optimized pa-rameters for all four mathematical models (Eqs. (1)–(2)) and all threedensity groups can be seen in Table 4. R2 measures how successful thefit is in explaining the variation of the data i.e. R2 is the correlationbetween predicted and observed values. The parameter R2 was 0.990or higher in all fits. Based on the experimental data which can be seenin Fig. 7 (triangles) we can see that the percentage of electroporatedcells depends significantly on the cell density. In Fig. 7 data is presentedonly with Gompertz curve—the reason is explained later in this section.Somewhat surprising, the mathematical models of electroporation arenot shifting to the higher values of E but are changing their slopeswith different densities of the monolayer. All of the curves start toincrease at approximately the same point (0.4 kV/cm) but reach theirplateaus at different values of the electric field (1.2 kV/cm or more).At lower values of the electric field (less than 0.4 kV/cm) there was noelectroporation detected. At the middle values of the electric field(0.4 kV/cm–1.2 kV/cm) where the curves are approximately linear,the slopes of thefittedmodels are different for each of the ranges of den-sities. When the density of the cells was lower the curve was steeper.We can see the change in the slope in Fig. 7 where the Gompertzcurve (Eq. (3)) was fitted to all three density groups.

For further analysis images with the density of the cells in the rangefrom 300 to 600 cells per image were chosen. In the experiments in theinhomogeneous electric field the actual density of the cells was higherin some areas of the composed image (more than 600 cells per image)but lower in the others. Therefore, 300 to 600 cells per image wereselected as an approximation for an average cell density and furtheranalysis was based only on density from 300 to 600 cells per image.

Fig. 8 presents the influence of the chosen mathematical model onthe predicted percentage of electroporated cells. It seems as if therewere only two different models shown instead of four. Namely, hyper-bolic tangent and symmetric sigmoid are completely overlapping andtherefore there is no visible difference on the graph. The same is truefor the Gompertz curve and asymmetric sigmoid. All the mathematicalmodels were used with parameters fitted to data from experiments inthe homogeneous field. From the R2 coefficients in Table 4 we canobserve that the Gompertz curve and asymmetric sigmoid when thedensity of the cells is 300–600 cells per image offer the best fit to the ex-perimental data of the four curves used. Although Gompertz curve andasymmetric sigmoid are overlapping and both offer almost equally goodfit to the experimental datawe have chosen the Gompertz curve (Eq. (3))for further analysis.

In Fig. 7 we can observe the influence of the density of themonolay-er, whereas in Fig. 8 we can see the influence of the chosen mathemat-ical model on the percentage of electroporation. Fig. 9 is based on Figs. 7and 8, as it combines the appropriate density of the monolayer (Fig. 7)and the best fit based on the R2 value (Fig. 8). The appropriate densityis the density of the monolayer exposed to an inhomogeneous field,i.e. 300–600 cells per image.

3.2. Cell electroporation in an inhomogeneous electric field

On the basis of the results acquired in experiments in the homoge-neous electric field and the model of geometry of the two needleelectrodes different mathematical models of cell electroporation as afunction of electric field (Eqs. (1)–(4)) were applied to the numericallycalculated inhomogeneous electric field.

We transformed a numerically calculated electric field (Fig. 5a) intothe predicted percentage of the electroporated cells (Fig. 6b). In Fig. 9

Fig. 5.Distribution of the electric field strength in the plane of cell monolayer. a—Continuousdistribution of the electric field strengthwhen 120 V is applied. b—Contours whichmark theborders between different ranges of the electric fieldwhen 120 V is applied. Range of electricfield in a certain area is written in the corresponding area. c—Fluorescent microscopic com-posed image with superimposed contours of ranges of electric field, one 1 ms pulse of 120V applied.

image of Fig.�5

p(%)

b

a

Fig. 6. a—Thresholded fluorescent microscopic image, averaged 6 images, approximateposition of the electrodes is marked with gray circles; signs + and − in the circles markthe polarity of the electrodes. b—Predicted percentage of the electroporated cells (p) inthe plane of cell monolayer when 120 V is applied; transformation from the electricfield strength to the predicted percentage is made by the Gompertz curve for densitiesfrom 300 to 600 cells per image.

Table 3Ranges of E-field vector when different voltages are applied and the size of the area in percents of the whole analyzed area without the electrodes. If a certain range is not analyzed whenthat voltage is applied there is a sign – in the corresponding cell. When the sum of percentages is not 100% it is so because of rounding of the numbers.

E-field strength range (kV/cm) 80 V area (%) 100 V area (%) 120 V area (%) 140 V area (%)

0.3–0.4 33 30 29 280.4–0.6 47 33 29 280.6–0.8 12 19 16 150.8–1.0 – 12 13 100.8–1.5 8 – – –

1.0–1.2 – – 7 101.0–1.6 – 6 – –

1.2–2.0 – – 5 –

1.2–2.3 – – – 9



we can see both the theoretically predicted and experimentally deter-mined values. With the variation of the different models used formodeling the phenomena the difference in predicted ranges was mini-mal. This can be seen from comparing black vertical bars (Gompertzcurve (Eq. (3)), 300 to 600 cells per image) and dark gray vertical bars(symmetric sigmoid (Eq. (1)), 300 to 600 cells per image) in Fig. 9.The predicted ranges of the percentage of the electroporated cells arealmost the same. The difference in ranges was maximally 4% at lowerelectric field strengths. If we compare ranges predicted by these twocurves (Eqs. (1) and (3)) based on the same density (300 to 600 cellsper image) with the light gray bars, which represent a densermonolay-er (more than 600 cells per image), we can observe that the predictedranges are quite different at lower as well as at higher electric fieldstrengths. Other combinations of the interpolation curve and the densi-ty of the cells were made as well (Table 4) but for the sake of stressingthe influence of the density and of the mathematical model only theGompertz curve for 300–600 and more than 600 cells per image andsymmetric sigmoid for 300–600 cells per image are shown. Aside fromthe symmetric sigmoid (Eq. (1)) also the hyperbolic tangent (Eq. (4))could be shown.

4. Discussion

The aim of our studywas to compare differentmathematical modelsthat would allow transformation of the numerically calculated values ofthe electric field into the predicted percentage of electroporated cells.This kind of transformation and prediction would simplify presentationof treatment plans for electrochemotherapy and non-thermal irrevers-ible electroporation. We upgraded the usual assumption that the per-centage of the electroporated cells is 100% if the electric field is abovethe characteristic threshold and 0% if it is below [49]. Here the predic-tion was continuous and all the values between 0% and 100% were pre-dicted.We investigated and compared the effects of the cell density andof the used mathematical model (Eqs. (1)–(4)). We started our studywith the model of continuous electric field distribution presented inFig. 5a, transforming it into the predicted percentage of electroporatedcells as shown in Fig. 6b. In Fig. 6a we can see that the pattern of theelectroporated and non-electroporated cells is in good agreementwith the predicted shape in Fig. 6b.

Ifwe look at Fig. 6a it appears as if thereweremore cells electroporatedaround the positive electrode. However, the analysis of the percentages ofelectroporated cells around each of the electrode (data not shown)showed that there was no significant difference between the percentagesaround the positive and around the negative electrode.

In the course of the transformation from the electric field strength inthe homogeneous field to the percentage of electroporated cells in theinhomogeneous electric field we reached two main conclusions. Thefirst one was about the changing of the slope of the mathematicalmodel of electroporation in the homogeneous field and the secondone was about the choice of the mathematical model, fitted to the

55

Table 4Results for the goodness of thefit (R2) and the curves' parameters for all of the four proposed curves for all three density groups. The fit is based on the percentage of electroporated cells inthe homogenous field. The parameters are explained in 2.4 Fitting of the mathematical models.

Type of the curve R-square a Parameters

b300 cells/image 300–600 cells/image N600 cells/image b300 cells/image 300–600 cells/image N600 cells/image

Symmetric sigmoid 0.992 0.990 0.964 E50% = 0.659 E50% = 0.6879 E50% = 0.9231b = 0.1090 b = 0.1242 b = 0.2353

Asymmetric sigmoid 0.997 0.998 0.965 E50% = 0.5869 E50% = 0.6061 E50% = 1.057v = 5e−8 (fixed at bound) v = 2e−8 (fixed at bound) v = 2.557b = 0.1529 b = 0.1772 b = 0.1494

Gompertz curve 0.997 0.998 0.958 E50% = 0.5869 E50% = 0.6061 E50% = 0.7567b = 0.1529 b = 0.1772 b = 0.343

Hyperbolic tangent 0.992 0.990 0.964 E50% = 0.659 E50% = 0.6879 E50% = 0.9231B = 4.587 B = 4.027 B = 0.9231

a Goodness-of-fit.



experimental data. In general results obtained are in good agreementwith the results found in the literature, describing experiments withsuspensions. Nevertheless, the change of the models' slopes with re-spect to cell density seems rather surprising and contradictory to theo-retical considerations [47].

First, we will discuss the change of models' slope. Experimental ob-servationwhere the slope of themodel changes can be observed in [35].There we can see that the curve of detected fluorescence shifted with achange of pulse parameters (when longer pulses were used the slopewas steeper). No curve which would show the dependence on the celldensity is shown; we can only see that with the same pulse protocoland higher density of the suspension fewer cells are electroporated.

As it can be observed from Fig. 7, the model of electroporation didnot shift but changed its slope when monolayers of cells with differentdensities were used for experiments. In previous studies it has been ob-served that with increasing densities of cell suspensions the base pointof the curves and the pointwhere the curves reach their plateaus shiftedto the higher electric field with the slope of the curve being the same[47]. In our study one part of the observation was similar—the pointswhere the curves reach their plateau values shifted to the higher electricfield valueswhenwe increased the density of the cellmonolayer. On theother hand, the base point where a minimal fluorescence of the cells

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

0

10

20

30

40

50

60

70

80

90

100

E(appl) / (kV/cm)

p / %

Gompertz curve applied to all three density groups

Up to 300 cells per image300 to 600 cells per imageAbove 600 cells per imageUp to 300 cells per image300 to 600 cells per imageAbove 600 cells per image

Fig. 7. Percentage (p) of electroporated cells determined by propidium iodide staining inthe dependence on the applied electric field. The Gompertz curve is fitted to all threegroups of different densities of the cells. Mean experimental values from experiments,made in the homogeneous field are marked with triangles; vertical bar denotes one stan-dard deviation. Black triangle and gray solid line represent density up to 300 cells perimage, hollow triangle and dotted black line represent density from 300 to 600 cells perimage, and gray triangle and dashed black line represent density above 600 cells perimage.

56

was detected was the same for all of the densities. Therefore, the slopeof the curves changed.

It is known that with denser suspensions we get lower inducedtransmembrane voltage due to the mutual electrical shielding[33–36]. In the monolayers the situation is the same—with densermonolayers we get lower induced transmembrane voltage andlower percentage of the electroporated cells. However, we shouldnot neglect the effect of the cells' geometry [50,51] which is deviato-ry from spheres and the electrical connections between the cells, e.g.gap junctions [52]. It seems that the cell's geometry and connectionsbetween the cells are related to the curves' slopes, whichmight be anarea of further research.

The standard deviation in Figs. 7 and 8 is relatively high; the reason iscounting of the cells. Allmeans of cell counting are subjected to errors be-cause of noise and artifacts, various cell shapes, and cells in close contactwithout clear boundaries between them. Considering that we hadmono-layers of very high density the calculated standard deviation is within theexpected values as reported in the literature [53]. The error would belower if using cells in suspension; however the cells in suspension andin tissues behave very differently. In a suspension there are no connec-tions between the cells and they are all approximately spherical.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

0

10

20

30

40

50

60

70

80

90

100

E(appl) / (kV/cm)

p / %

Applied Gompertz curve, hyperbolic tangent, symmetric sigmoid, asymmetric sigmoid

Gompertz curveAsymmetric sigmoidHyperbolic tangentSymmetric sigmoid300 to 600 cells per image

Fig. 8. Percentage (p) of electroporated cells determined by propidium iodide staining independence on the applied electric field. Four different models of electroporation werefitted to the experimental data of the percentage of electroporated cells in the homoge-neous field. Triangles represent the mean experimental values for the percentage ofelectroporated cells in the homogeneous electric field; vertical bar denotes one standarddeviation. Black solid line represents Gompertz line, black dashed line represents asym-metric sigmoid, gray solid line represents hyperbolic tangent and gray dashed line repre-sents symmetric sigmoid.

0

10

20

30

40

50

60

70

80

90

100

U / V : E(appl) / (kV/cm)

p / %

80: 0.30 − 0.39

100: 0.30 − 0.39

120: 0.30 − 0.39

140: 0.30 − 0.39

80: 0.40 − 0.59

100: 0.40 − 0.59

120: 0.40 − 0.59

140: 0.40 − 0.59

80: 0.60 − 0.79

100: 0.60 − 0.79

120: 0.60 − 0.79

140: 0.60 − 0.79

80: 0.80 − 1.50

100: 0.80 − 0.99

120: 0.80 − 0.99

140: 0.80 − 0.99

100: 1.00 − 1.60

120: 1.00 − 1.19

140: 1.00 − 1.19

120: 1.20 − 1.99

140: 1.20 − 2.30

Gompertz curve, from 300 to 600 cells/imageSymmetric sigmoid, from 300 to 600 cells/imageGompertz curve, more than 600 cells/image

Fig. 9. Predicted (vertical bars) and experimentally determined percentage (triangles) of the electroporation; model for a cell response made by the Gompertz curve (Eq. (3)) and sym-metric sigmoid (Eq. (1)) for different densities. Three bars in one set from left to right are the Gompertz curve for densities from 300 to 600 cells per image (black bar), the symmetricsigmoid for densities from 300 to 600 cells per image (dark gray bar) and the last bar the Gompertz curve for densities of more than 600 cells per image (light gray bar). The verticalbar at experimentally determined values represents one standard deviation. Labels on the x-axis mean the applied voltage and the area with the corresponding electric field range forthat applied voltage.



In the past, differentmathematicalmodels havebeenproposed for de-scribing the dependence between the percentage of the electroporatedcells and the electric field. For example, a hyperbolic tangent law forpermeabilization as a function of the electric field has been proposed[47]. In statistical physics the hyperbolic tangent is commonly used fordescribing two-state systems, for example the polarized light. Simi-larly, electroporated and non-electroporated cells can also be viewedas two states in a system and hyperbolic tangent law describes thecrossover from the non-electroporated to the electroporated state.

In [20] two different ways of describing the cell electroporation frac-tion have been used. The first one was sigmoid function which is oftenfitted to the experimental data. The secondonewas a curvewhichderivedfrom a hypothetical normal distribution of cell radii. In this case the curvewas obtained from a step function (electroporation is 0% when the ap-plied electric field is under the threshold for electroporation and 100%when the applied electric field is above the threshold). Cell radius wasvaried according to the normal distribution with empirically determinedvalues formean cell radius and its standard deviation. Goodness-of-fit be-tween the normal distribution curve on one side and the experiments onthe other showed that experimental resultswere in good agreementwiththe theoretically predicted values. Nevertheless, the authors could not de-cide which of the two curves was more appropriate.

The Gompertz curve is used for describing the systems which satu-rate in a long period of time, for example the growth of the tumors[46]. The growth is slower at the beginning. Then the size starts to in-crease faster. The growth is limited after a certain time period whenthe size of the tumor reaches a plateau value. The growth can be com-pared to the percentage of cell electroporation as a function of the elec-tric field where we obtain high percentage of cells being electroporated,butwith an increasing electric field the increase e.g. from 97% to 100% ofelectroporated cells is difficult to obtain.

Up to a certain electric field reached there are almost no cellselectroporated. From 0.4 kV/cm to 1.0 kV/cm the percentage of theelectroporated cells quickly increases and then it reaches the plateauvalue. The described logic is the reason that we proposed the Gompertzcurve for describing the electroporation of the cells. It is not necessary

that the Gompertz curve is symmetric. The asymmetry in the model isappropriate because in reality the percentage of the electroporatedcells depends also on the cell radius, and according to the literaturethe cell radii are not symmetrically distributed [20] which was alsothe case in our study (data not shown).

Therefore, we can expect that themathematical model of electropo-ration is asymmetric as well. This asymmetry was also the reason whywe chose the asymmetric sigmoid curve for analysis.

So far not many studies of a statistical evaluation of electroporationare available. For evaluating the area of irreversible electroporation astatistical model based on the Peleg–Fermi model combined with anumerical solution of the multidimensional electric field equation castin a dimensionless formwas used [21]. This model directly incorporatesthe dependence of cell death on pulse number (n) and on electric field(E). It is expressed by Eqs. (5)–(7), where Smeans the survival ratio andEc marks the intersection of the curve with the y-axis. Coefficients k1and k2 are cell type and pulse type specific.

S ¼ 1= 1þ exp E−Ec nð Þð Þ=A nð Þð Þ ð5Þ

Ec nð Þ ¼ Ec0� exp −k1

�n� � ð6Þ

A nð Þ ¼ A0� exp −k2

�n� � ð7Þ

The problemwith this model is that it was not validated since it wastested only on extrapolated data reported in the literature for prostatecancer cell death caused by irreversible electroporation [54]. Authorsstated that real curves and parameters should be developed for eachspecific tissue. Also the Fermi–Peleg model should be validated firstin vitro and then in vivo.

Severalmicrobial inactivation curves have been effectively describedby Weibull distribution. In this model parameters were dependent onthe media type and treatment parameters (electric field and treatmenttime) [22,55] but not on pulse number and pulse length like thePeleg–Fermi statisticalmodel. Therefore, thismodel is not as interesting

57



for our study as the Peleg–Fermimodel. Also, in the area ofmicrobial in-activation by pulsed electric fieldsmanymathematicalmodels exist [22,56], but they all describe survival of the cells in dependence on appliedelectric field and treatment time, with treatment time most oftenreported as the sum of duration of all of the applied pulses. All thesemodels (Weibull, Peleg–Fermi, log-linear etc.) describe the survivalof the cells. In our study on the other hand, survival of the cells hasnot been determined. Therefore, these models were not used in ourstudy.

In our study four mathematical models (Eqs. (1)–(4)) were chosen,used and evaluated.We achieved good agreementwith all of them sinceR2 was 0.990 or higher in all four cases (see Table 4); therefore, for theanalysis of the effect of the cell density on electroporation any of themmight be used. For further analysis of the effect of cell density we con-sidered the two curves with the highest R2—Gompertz curve and asym-metric sigmoid. If parameter v in the asymmetric sigmoid model(Eq. (2)) was negative, no fit could be achieved because complex valueswere computed by model function. Therefore, we set a lower limit forthis parameter at 0. Although we managed to complete the fitting, theparameter v was fixed at bound, which meant that the best fit was notachieved. The asymmetric sigmoid model was thus not used in thenext step of the analysis.

For the analysis of the effect of the interpolation curve on the predic-tion of the percentage of electroporation we used only the Gompertzcurve (Eq. (3)) and symmetric sigmoid (Eq. (1)). There was no needto do the analysis bothwith hyperbolic tangent (Eq. (4)) and symmetricsigmoid (Eq. (1)) since they can be seen as equivalent (see their over-lapping in Fig. 8).

If we look at Fig. 9 we can see that under our experimental condi-tions the percentage of the affected cells depends more on the densityof the cells than on a type of the curve. When different curve wasused for the same density (compare dark gray bars for the symmetricsigmoid (Eq. (1)) and black bars for the Gompertz curve (Eq. (3)) inFig. 9) the difference between predicted ranges was 4% at lower electricfield strengths (0.30–0.39 kV/cm) and even less for the higher ones. Thereason for the 4% difference can be observed from Fig. 8. There we cansee that at lower electric field values Eqs. (1) and (3) deviate the mostone from another.

At lower electric field values the symmetric sigmoid overestimatesthe experimental results while the Gompertz curve offers better fit.This means that the percentage predicted by the symmetric sigmoid(Eq. (1)) is higher than the one predicted by the Gompertz curve(Eq. (3)) which is not in very good agreement with experimentalresults. But since the predicted ranges of electroporation are still quitesimilar (0–4% for Eq. (3) and 4–8% for Eq. (1)) and they both underesti-mate experimental results we can conclude that the choice of the curveis not of highest importance.

The reason for discrepancy at the higher electric field could be thefact that mathematical models for electroporation allow 100% electro-poration although in reality there are always some cells which do notrespond to electric pulses and stay unaffected at least to very highvalues of electric field. This is particularly true with single pulse appliedat very high electric fields (data not shown) as was the case in ourexperiments. This could be the reason why the predicted ranges inFig. 9 are above the experimentally measured values.

In vitro a small fraction of dead cells is always detected, which ex-plains the deviation of experimental data at the lowest electric fieldstrength from theoretical prediction by mathematical models forelectroporation.

If we look at the light gray bars at Fig. 9 (Gompertz curve (Eq. (3)) fordensities above 600 cells) per image, we can see that they do not repro-duce the experimental results (triangles) properly. The mathematicalmodel of electroporation is underestimating the experimental resultsat all of the applied electric field strengths for at least 10%. The reasonis in the density of the cells. Prediction was made on monolayers ofmore than 600 cells per image. The experiments were performed

58

on monolayers of less than 600 cells per image. This means that thedensity of the cells is a very important factor. It must be the samein experiments used for prediction and in experiments where wepredict the percentage of electroporated cells. The prediction offeredbetter agreement only at the higher values of the electric field be-cause the electric field was already strong enough for all of the curvesto reach their plateau values. Therefore, we can say that the densityof the cell monolayer is very important for predicting the percentageof electroporation.

In previously published works a strong dependence between thecells' electroporation and the density of the cells was already shown[33,34]. In our study we went one step further and showed that thecell density not only has a strong influence on the cells' electroporationbut is under our experimental conditions the most important factorinfluencing the prediction of electroporation. We eliminated the effectof the size of the cells since the experiments in homogeneous and inho-mogeneous field were made on monolayers of similar density. Ourexperiments have been performed in vitro on CHO cells. Our resultswere obtained using single pulse of 1 ms duration; however we needto establish how the parameters of curves depend on duration andnumber of pulses, and different cells. In addition, there might be otherparameters besides the density of the cells which have an important in-fluence on the prediction of electroporation.How this translates into tis-sue remains to be determined; tissue level determination and validationare still needed [38,39].

Acknowledgments

This study was supported by the Slovenian Research Agency (ARRS)and conducted within the scope of the Electroporation in Biology andMedicine European Associated Laboratory (LEA-EBAM). Authors (J.D.)would like to acknowledge doc. dr. Gorazd Pucihar and Duša Hodžićfor their guidance when starting to work in cell laboratory. Experimen-tal work was performed in the infrastructure center ‘Cellular ElectricalEngineering’.

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60


Paper 2 Title: Mathematical Models Describing Chinese Hamster Ovary Cell Death Due to Electroporation In

Vitro


Publication: Journal of Membrane Biology, vol. 248, no. 5, pp. 865–881, Oct. 2015.

DOI: 10.1007/s00232-015-9825-6


Quartile: Q3 (Biochemistry & Molecular Biology, Physiology)

Rank: 207/289 (Biochemistry & Molecular Biology), 43/83 (Physiology)

61

Mathematical Models Describing Chinese Hamster Ovary CellDeath Due to Electroporation In Vitro

Janja Dermol1 • Damijan Miklavcic1

Received: 21 January 2015 / Accepted: 16 July 2015 / Published online: 30 July 2015

� Springer Science+Business Media New York 2015

Abstract Electroporation is a phenomenon used in the

treatment of tumors by electrochemotherapy, non-thermal

ablation with irreversible electroporation, and gene therapy.

When treating patients, either predefined or variable elec-

trode geometry is used. Optimal pulse parameters are pre-

determined for predefined electrode geometry, while they

must be calculated for each specific case for variable elec-

trode geometry. The position and number of electrodes are

also determined for each patient. It is currently assumed that

above a certain experimentally determined value of electric

field, all cells are permeabilized/destroyed and under it they

are unaffected. In this paper, mathematical models of sur-

vival in which the probability of cell death is continuously

distributed from 0 to 100 % are proposed and evaluated.

Experiments were performed on cell suspensions using

electrical parameters similar to standard electrochemother-

apy and irreversible electroporation parameters. The pro-

portion of surviving cells was determined using clonogenic

assay for assessing the ability of a cell to grow into a colony.

Various mathematical models (first-order kinetics, Hul-

sheger, Peleg-Fermi, Weibull, logistic, adapted Gompertz,

Geeraerd) were fitted to experimental data using a non-linear

least-squares method. The fit was evaluated by calculating

goodness of fit and by observing the trend of values of

models’ parameters. The most appropriate models of cell

survival as a function of treatment time were the adapted

Gompertz and the Geeraerd models and, as a function of the

electric field, the logistic, adapted Gompertz and Peleg-

Fermi models. The next steps to be performed are validation

of the most appropriate models on tissues and determination

of the models’ predictive power.

Keywords Clonogenic assay � Cell death probability �Treatment planning � Electrochemotherapy � Predictivemodels � Non-thermal irreversible electroporation

Introduction

Electroporation is a phenomenon that occurs when short

high voltage pulses are applied to cells and tissues. This

exposure of cells to electric pulses results in pores being

formed in the cell membrane. Membranes become per-

meable to molecules that cannot otherwise pass in or out of

the cell (Kotnik et al. 2012; Weaver 1993). If the cell is

able to recover, it is considered reversible electroporation.

If the damage to the cell is too extensive and the cell dies, it

is considered irreversible electroporation. The existence of

the pores has been shown by molecular dynamics (Dele-

motte and Tarek 2012) and calculated by various theoret-

ical models (Neu and Neu 2009; Weaver and Chizmadzhev

1996). Electroporation is already being used in medicine,

e.g., electrochemotherapy (Edhemovic et al. 2014), non-

thermal irreversible electroporation as a method of tissue

ablation (Cannon et al. 2013; Davalos et al. 2005; Garcia

et al. 2014; Long et al. 2014; Neal et al. 2013), gene

therapy (Daud et al. 2008; Heller and Heller 2010), DNA

vaccination (Calvet et al. 2014), and transdermal drug

delivery (Denet et al. 2004; Yarmush et al. 2014), as well

as in biotechnology (Kotnik et al. 2015) and food pro-

cessing (Mahnic-Kalamiza et al. 2014; Sack et al. 2010). It

has been shown that a sufficient electric field (E-field) is

the most important factor—all the cells in the tumor have

to be permeabilized (in electrochemotherapy) (Miklavcic

& Damijan Miklavcic

[email protected]

1 Faculty of Electrical Engineering, University of Ljubljana,

Trzaska 25, 1000 Ljubljana, Slovenia

123

J Membrane Biol (2015) 248:865–881

DOI 10.1007/s00232-015-9825-6


62

http://crossmark.crossref.org/dialog/?doi=10.1007/s00232-015-9825-6&domain=pdf

http://crossmark.crossref.org/dialog/?doi=10.1007/s00232-015-9825-6&domain=pdf

et al. 1998) or irreversibly electroporated (in irreversible

electroporation) to eradicate the tumor. E-field distribution

also corresponds to tissue necrosis (Long et al. 2014;

Miklavcic et al. 2000).

When performing electrochemotherapy, irreversible

electroporation or gene therapy, fixed electrode configu-

rations with predefined pulse parameters can be used

(Heller et al. 2010; Mir et al. 2006). Alternatively, variable

electrode configurations can be used when the target tumor

is outside the standard parameters (Linnert et al. 2012;

Miklavcic et al. 2012). When using variable electrode

configurations, a plan is needed of the electrodes’ position

and the parameters of electric pulses that offer sufficient E-

field in the tissue (Campana et al. 2013; Miklavcic et al.

2010; Neal et al. 2015; Sel et al. 2007; Zupanic et al. 2012).

Treatment planning of electroporation-based medical

applications has already been successfully used on col-

orectal liver metastases in humans (Edhemovic et al. 2014),

and on spontaneous malignant intracranial glioma in dogs

(Garcia et al. 2011a, b). It is currently assumed in treatment

plans that above an experimentally determined threshold

value of E-field, all cells are permeabilized or destroyed

and below this threshold cells are not affected or do not

die—i.e., we assume a step-like response. In reality,

though, the transition from non-electroporated to electro-

porated state and from reversibly to irreversibly electro-

porated state is continuous. Mathematical models of cell

permeabilization and survival can be implemented in order

to present treatment plan in a clearer way and to obtain a

better prediction of the tissue damaged (Dermol and Mik-

lavcic 2014; Garcia et al. 2014). In addition, mathematical

models allow us to interpolate the predicted survival of the

cells and predict survival for other parameters than those

used for curve fitting. The mathematical models of survival

have to be adaptable and describe the experimental data

well (high goodness of fit). Goodness of fit is not only an

important criterion but trends of the optimized values of the

parameters and the predictive power of the model are also

important. In an ideal case, the models would include all

the parameters important for cell death due to electropo-

ration, but would have the lowest possible number of

parameters.

There are only a few reports describing the probability

of cell permeabilization (Dermol and Miklavcic 2014) and

cell survival after irreversible electroporation (Garcia et al.

2014; Golberg and Rubinsky 2010) using mathematical

models. The first attempt using mathematical model of

survival to describe cell death after irreversible electropo-

ration was made by (Golberg and Rubinsky 2010). They

successfully fitted the Peleg-Fermi model to experimental

data of prostate cancer cells’ death described in (Canatella

et al. 2001). Later, (Garcia et al. 2014) simulated irre-

versible electroporation on liver tissue and characterized

cell death using the Arrhenius rate equation for thermal

injury and the Peleg-Fermi model for electrical injury. The

authors determined that using commercially available

bipolar electrodes (AngioDynamics, Queensbury, USA)

and standard irreversible electroporation parameters (90

pulses, 100 ls duration, 1 Hz, 3000 V) most cell death is a

consequence of electrical damage. In that study, the vol-

ume of the thermally destroyed tissue did not surpass 6 %

of the whole destroyed volume and was concentrated in the

immediate vicinity of the electrodes.

Until now, the Peleg-Fermi model has been the only

mathematical model used for describing cell death as a

consequence of irreversible electroporation in medicine.

However, mathematical modeling of cell death has a long

history in the field of microbiology, e.g., food sterilization

(Peleg 2006). Most models from the field of microbiology

describe thermal microbial inactivation; an independent

variable is treatment time (t). We used these models (first-

order kinetics, Weibull, logistic, adapted Gompertz, Geer-

aerd) in original and in transformed forms. In the original

forms, the models remained unchanged, treatment time was

an independent variable, and E-field was a parameter. In

the transformed forms, E-field became the independent

variable. There was no need to transform the Hulsheger

model and the Peleg-Fermi model, since the independent

variables were E-field and treatment time (Hulsheger et al.

1981) or E-field and the number of pulses (Peleg 1995). In

existing studies, mathematical models have not yet been

used as a function of the E-field. Since E-field is a domi-

nant parameter for predicting the effect of the electropo-

ration, we were interested in obtaining models as a function

of E-field. We also provide an explanation of the reasoning

for the transformation for each of the transformed models.

In (Canatella et al. 2001), the authors exposed prostate

cancer cells to 1–10 exponentially decaying pulses in the

range of 0.1–3.3 kV/cm, with time constants in the range of

50 ls–20 ms. In our experiments, up to 90 square pulses,

0–4.0 kV/cm, 50–200 ls were applied, which are typically

used in electrochemotherapy and irreversible electropora-

tion treatments. Clonogenic assay was used as a measure of

the ability of the cells to reproduce (Franken et al. 2006).

Our study is the first attempt to compare different

mathematical models describing the survival of animal

cells due to electroporation. We present the results

obtained with electrical parameters similar to those typi-

cally used in electrochemotherapy and in irreversible

electroporation clinical treatments. We evaluate the trends/

meaning of the parameters of the mathematical models,

determine whether and which models could be used for

describing cell death, and which models should be vali-

dated in the treatment planning and treatment response

prognostics of electrochemotherapy and irreversible

electroporation.

866 J. Dermol, D. Miklavcic: Mathematical Models Describing Chinese Hamster Ovary Cell Death…

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63

Materials and Methods

Cell Preparation and Electroporation

Chinesehamster ovary cells (CHO-K1;EuropeanCollectionof

Cell Cultures, Great Britain) were grown in 25-mm2 culture

flasks (TPP, Switzerland) for 2–3 days in an incubator (Kam-

bic, Slovenia) at 37 �C and humidified 5 % CO2 in HAM-F12

growth media (PAA, Austria) supplemented with 10 % fetal

bovine serum (Sigma Aldrich, Germany), L-glutamine (Stem-

Cell, Canada) and antibiotics penicillin/streptomycin (PAA,

Austria), and gentamycin (Sigma Aldrich, Germany). The cell

suspensionwas prepared on the day of experiments. Cells were

centrifuged and resuspended in potassium phosphate electro-

poration buffer (10 mM K2HPO4/KH2PO4 in a ratio 40.5:9.5,

1 mM MgCl2, 250 mM sucrose, pH 7.4, 1.62 mS/cm,

260 mOsm) at a concentration 106 cells/ml.

A drop of cell suspension (100 ll) was pipetted between

two parallel stainless steel electrodes (Fig. 1) with the dis-

tance between them set at 2 mm. The surface of the elec-

trodes was much larger than the contact surface between the

cell suspension and the electrodes. All the cells were thus

exposed to approximately the same electric field, which was

estimated as the voltage applied divided by the distance

between the electrodes. Pulses were delivered using a

Betatech electroporator (Electro cell B10 HVLV, Betatech,

France) and monitored with an oscilloscope LeCroy Wave-

Surfer 422, 200 MHz and a current probe AP015 (both

LeCroy, USA). The parameters of the applied electric pulses

are summarized inTable 1. The electrodeswerewashedwith

sterile 0.9 % NaCl and dried with sterile gauze between

samples. After pulse application, 80 ll of cell suspensionwas transferred into a microcentrifuge tube in which there

was already 920 ll of HAM-F12. In the control, sample cells

were put between the electrodes and no pulses were deliv-

ered. Control was performed at the beginning and at the end

of each experiment to monitor whether the survival or

number of cells in the suspension during an experiment had

decreased. After all the samples had been exposed to electric

pulses (10–20 min), cells were diluted in 0.9 % NaCl and

plated in triplicates in 6-well plates (TPP, Switzerland) in

3 ml of HAM-F12. Different numbers of cells were plated,

shown in Table 2 for parameters with different lengths of

pulses and in Table 3 for parameters with different numbers

of pulses (Franken et al. 2006). In preliminary experiments,

we determined how many cells have to be seeded in order to

obtain around 100 colonies per well. When more intense

treatments (600 V, 50–90 pulses and 800 V, 30–90 pulses)

were applied, often no cells survived but, because of

experimental conditions, we could not seed more cells than

the given number. Cells were grown for 6 days at 37 �C and

5 % CO2.

After 6 days, HAM-F12 was removed and cells were

fixed with 1 ml of 70 % ethanol (Lekarna Ljubljana,

Slovenia) per well. Cells were left in ethanol for at least

10 min,which rendered all cells dead. Colonieswere colored

with 200 ll of crystal violet (0.5 %w/v in distilledwater) per

well. Excessive crystal violet was washed away with pipe

water. Colonies that had more than 50 cells were counted.

The proportion of surviving cells (S) was calculated as

S ¼ number of colonies after the treatment

number of seeded cells � PE; ð1Þ

where plating efficiency (PE) is defined as

PE ¼ number of colonies formed in the control

number of colonies seeded in the control: ð2Þ

The number of colonies formed in the control was cal-

culated as an average of the triplicates of the colonies formed

in the controls at the beginning and at the end of the exper-

iment. The number of colonies seeded in control samples was

100. There was no difference between the two controls, so we

could pool the results. At least four independent experiments

were performed for each pulse parameter, and mean and

standard deviation were then calculated.

In order to determine whether our pulses were indeed

not causing significant heating, we measured the temper-

ature of the cell suspension before any pulses were applied

and within 5 s after the application of 90 pulses, 4.0 kV/

cm, i.e., the most intense exposure. Because of the limi-

tations of the temperature probe, the temperature could not

be measured during the pulse application. We used the

fiber optic sensor system ProSens (opSens, Canada) with a

Fig. 1 Stainless steel parallel plate electrodes with a droplet of cell

suspension between the electrodes (upper image). Inter-electrode

distance is 2 mm

J. Dermol, D. Miklavcic: Mathematical Models Describing Chinese Hamster Ovary Cell Death… 867

123


64

fiber optic temperature sensor, which was inserted in the

cell suspension between the electrodes. In addition, a

numerical model of the cell suspension droplet between the

electrodes was made and temperature distribution after

90 pulses of 4.0 kV/cm was calculated (Appendix).

Fitting of the Mathematical Models of Survival

to the Experimental Results

Different mathematical models of survival were fitted to the

experimental data: (i) the first-order kinetics model (Bige-

low 1921), (ii) the Hulsheger model (Hulsheger et al. 1981),

(iii) the Peleg-Fermi model (Peleg 1995), (iv) the Weibull

model (van Boekel 2002), (v) the logistic model (Cole et al.

1993), (vi) the adapted Gompertz model (Linton, 1994), and

(vii) the Geeraerd model (Geeraerd et al. 2000). The method

of non-linear least squares was applied using Matlab

R2011b (Mathworks, USA) and Curve fitting toolbox.

Optimal values of the parameters of the mathematical

models and R2 values were determined. R2 or the coefficient

of determination is a statistical measure for the goodness of

fit, i.e., it is a correlation between the predicted and

experimentally determined values. Its values can be

between 0 and 1 and the closer its value is to 1, the better the

fit is. Natural logarithms of mathematical models were fitted

to natural logarithms of the experimental data. This pre-

vented the residuals at higher proportions of surviving cells

from influencing the R2 value the most. Treatment time t

was understood as the time of exposure of cells to the

electric field (E-field). It was calculated as

t ¼ NT ; ð3Þ

where N means number of the applied pulses and T the

duration of one pulse.

The first-order kinetics model has a long history

(Bigelow 1921):

SðtÞ ¼ expð�ktÞ: ð4Þ

Here, t denotes the time of exposure of bacteria to high

temperature and k is the first-order parameter, i.e., the

speed of decrease of the number of bacteria as a function of

the duration of their exposure to heat.

Hulsheger studied the effect of E-field on E. coli and

derived an exponential empirical model (Hulsheger et al.

1981):

Sðt;EÞ ¼ t

tc

� ��ðE�EcÞk

; ð5Þ

Table 1 Experimental

parameters of electric pulses,

pulse repetition frequency 1 Hz

Voltage/V Electric field/kV/cm Number of pulses/- Pulse duration/ls

Varying pulse length 0–800, step 100 0–4, step 0.5 8 50, 100 or 200

Varying pulse number 0–800, step 200 0–4, step 1 30, 50, 70 or 90 100

Table 2 Number of plated cells

in experiments with different

lengths of the pulses, 8 pulses,

1 Hz pulse repetition frequency

Voltage/V Electric field/kV/cm Number of plated cells/-

For 50 ls For 100 ls For 200 ls

0 0 100 100 100

100 0.5 100 100 100

200 1.0 120 120 120

300 1.5 150 150 150

400 2.0 200 200 200

500 2.5 200 400 400

600 3.0 400 1000 1000

700 3.5 1000 2500 2500

800 4.0 2500 10,000 25,000

Table 3 Number of plated cells

in experiments with different

numbers of the pulses, 100 ls,1 Hz pulse repetition frequency

Voltage/V Electric field/kV/cm Number of plated cells/-

For 30 pulses For 50 pulses For 70 pulses For 90 pulses

0 0 100 100 100 100

200 1.0 120 150 200 200

400 2.0 1000 2500 25,000 25,000

600 3.0 25,000 25,000 25,000 25,000

800 4.0 25,000 25,000 25,000 25,000


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65

where k is a constant, which depends on the type of

microorganism, Ec is the critical value of E-field below

which there will be no inactivation (100 % survival), and tcis the extrapolated critical value of t below which there will

also be no inactivation.

The Peleg-Fermi model (Peleg 1995) has already been

used for modeling irreversible electroporation (Garcia et al.

2014; Golberg and Rubinsky 2010) and is defined as

S E;Nð Þ ¼ 1

1þ expE�Ec Nð Þk Nð Þ

� � ; ð6Þ

EcðNÞ ¼ Ec0 expð�k1NÞ; ð7ÞkðNÞ ¼ k0 expð�k2NÞ; ð8Þ

where Ec means critical E-field, N is the number of applied

pulses, k is the kinetic constant that defines the slope of the

curve, Ec0 is the intersection of Ec(N) with the y axis, k0 is a

constant in kV/cm, k1 and k2 are non-dimensional constants

that depend on the parameters of the pulses and on the

cells.

The Weibull model describes the time to failure of

electronic devices after they have suffered some stress. The

Weibull model is based on the observation that cells die at

different times, which are statistically distributed. Cell

death due to electroporation can also be described using the

Weibull model (van Boekel 2002). We made a parallel: cell

death as a function of E-field is also statistically dis-

tributed. In addition, no-one has so far observed any cor-

relation between biological parameters and the parameters

of the Weibull model. We thus transformed the Weibull

model as a function of treatment time to a function of E-

field. In previous studies, the independent variable was

treatment time (time of exposure to high temperature or E-

field). The Weibull model as a function of t is

S tð Þ ¼ exp � t

b

� �n� �; ð9Þ

where t denotes time of exposure, b is a scale parameter,

and n is a shape parameter. We can transform the Weibull

model and obtain a model as a function of E:

S Eð Þ ¼ exp � E

b

� �n� �; ð10Þ

where all the parameters have the same meaning as in (9).

The logistic model can be used for describing distribu-

tions with a sharp peak and long tails (Cole et al. 1993).

The logistic model is defined as

S tð Þ ¼ 10

x�a

1þexp4rðs�log10ðtÞ

x�a

� � !

; ð11Þ

where parameter a denotes the common logarithm of the

upper asymptote (survival around t = 0), x the common

logarithm of the lower asymptote (survival when t ? ?),

r the maximum slope, and s the position of the maximum

slope. We measured the proportion of surviving cells. After

a short treatment time, most of the cells are still alive,

survival is 1. Therefore,

a ¼ log upper asymptoteð Þ ¼ log 1 ¼ 0: ð12Þ

This allows us to simplify the Eq. (11) by assuming

a = 0:

S tð Þ ¼ 10

x

1þexp4rðs�log10ðtÞ

x

� � !

; ð13Þ


A cumulative distribution of cell death was obtained in

the experimental results. This means that the experimental

data point of proportion of destroyed cells includes also

cells that would already die at shorter treatment times or

lower E-field values. A derivative of the cumulative cell

death distribution shows how cell death is spread over

different treatment times or E-field values; it shows cell

death distribution. Because our experimental data was

discontinuous, we obtained the derivative by calculating

the difference in survival between two consecutive data

points. We thus obtained the proportion of cells that die in

a certain range of treatment time or E-field values, for

example from 3000 to 5000 ls or from 1 to 2 kV/cm.

Shorter treatment times or lower E-field values do not kill

cells in that range. In dependence on the logarithm of the

treatment time, the derivative of the cumulative cell death

distribution (the derivative of experimental results) has a

sharp peak and two long tails. A similar shape of cell death

distribution is obtained as a function of E-field (without the

logarithm). As already mentioned, the logistic model is

suitable for distributions with a sharp peak and long tails.

This was our basis for the transformation from treatment

time as the independent variable to E-field as the inde-

pendent variable. The model is

S Eð Þ ¼ 10

x

1þexp4rðs�EÞ

xð Þ

� �; ð14Þ

where r and s have the same meaning as in (11) and xdenotes survival when E ? ?.

The Gompertz model is usually used for describing

growth of a tumor but in an adapted form it has also been

used for cell survival (Linton 1994):

S tð Þ ¼ exp Ae�e B0þB1 tð Þ � Ae�eB0� �

: ð15Þ

A denotes the natural logarithm of the lower asymptote, B0

is the length of the upper asymptote, and B1 is connected to

the speed of cell death. B1’s absolute value determines the

speed, the minus sign means a decrease in the number of


123


66

cells and plus means an increase. The adapted Gompertz

model is purely empirical and was chosen because it offers

an excellent goodness of fit. High R2 is also obtained if the

independent variable is E-field instead of treatment time.

This was the reason for the transformation from treatment

time as the independent variable to E-field as the inde-

pendent variable. We transformed the adapted Gompertz

model to a model as a function of E:

SðEÞ ¼ expðAe�e B0þB1Eð Þ � Ae�eB0 Þ; ð16Þ

where all the parameters mean the same as in (15).

Geeraerd defined a model that describes exponential

decay of the number of surviving cells and a lower

asymptote that models the remaining resistant cells

(Geeraerd et al. 2000; Santillana Farakos et al. 2013):

S tð Þ ¼ Y0 � Nresð Þexpð�ktÞ þ Nres: ð17Þ

Y0 means the number of cells at the beginning of experi-

ments, Nres is the lower asymptote, and k is the specific

inactivation rate (the slope of the exponentially decaying

part of the curve). In our study, the number of cells was

substituted by the proportion of cells in order to scale the

model to our experimental data. At the beginning of our

experiments, the proportion of survival was always 1. We

simplified the Geeraerd model into

S tð Þ ¼ 1� Nresð Þexpð�ktÞ þ Nres; ð18Þ


Results

Experimental Results

Figure 2 shows the experimental results—Fig. 2a for dif-

ferent pulse lengths and Fig. 2b for different numbers of

pulses. Experimental results are shown in a semi-loga-

rithmic scale, which enables low proportions of surviving

cells to be visualized. It can be observed in Fig. 2a that

longer pulses of the same electric field (E-field) cause

lower survival. However, with 8, 200 ls pulses of 4.0 kV/

cm, survival is still higher than with 50 or more 100 lspulses of 4.0 kV/cm (Fig. 2b). When the E-field increases

from 1.0 to 2.0 kV/cm, the survival after 50 pulses applied

drops by two decades, while after 70 and 90 pulses it drops

by three decades. In Fig. 2b, it can be seen that the

experimental values of survival are very similar for 50, 70,

and 90 pulses of 3.0 kV/cm and 4.0 kV/cm.

Results of the Mathematical Modeling

The results of the mathematical modeling are presented in

two ways. First, Tables 4, 5, 6, and 7 summarize the optimal

values of the parameters and R2 values for all the mathe-

matical models described in the Materials and Methods

section. Figures 3, 4, 5, and 6 show plotted optimized

mathematical models. When the results of the fitting are

presented in linear scale, the curves go straight and exactly

0 0.5 1 1.5 2 2.5 3 3.5 410-6

10-5

10-4

10-3

10-2

10-1

100

E /kV/cm

S/-

50 sµ100 sµ200 sµ

a

0 0.5 1 1.5 2 2.5 3 3.5 410-6

10-5

10-4

10-3

10-2

10-1

100

E /kV/cm

S/-

8 pulses30 pulses50 pulses70 pulses90 pulses

b

Fig. 2 Experimental results of the clonogenic assay for different

pulse lengths (a) and for different numbers of pulses (b) as a functionof the applied electric field (E). Mean ± one standard deviation is

shown, pulse repetition frequency 1 Hz. If mean minus standard

deviation is lower than 0, it cannot be presented in a semi-logarithmic

scale and there is no error bar. The survival is lower when longer

pulses are applied and the electric field is held fixed. However, higher

number of pulses decreases the survival considerably. With 8, 200 lspulses of 4 kV/cm, the survival is still higher than with 50 or more

100 ls pulses, 4 kV/cm


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67

through the experimental results. In semi-logarithmic scale,

deviations are more easily noticed. When fitting survival

models to experimental results, it is advisable also to look at

the data in semi-logarithmic scale. All our results are

therefore presented in semi-logarithmic scale. When com-

paring the values of our optimized parameters with the val-

ues in other studies, it must be borne inmind that on the x axis

there are t in ls or E-field in kV/cm.

Mathematical Models Describing Cell Survival

as a Function of Treatment Time

A good fit could not be achieved using the Hulsheger

model (5) because there were problems with the initial

value of the parameters and local minima. The results are

thus not shown and we do not discuss them.

In Fig. 3, (3.0 kV/cm, 100 ls, 1 Hz) it can be observed

that the Weibull model (9), the logistic (13), the adapted

Gompertz (15), and the Geeraerd models (18) are all

similarly shaped and go very close to the experimental

points. From the point of goodness of fit, they can be seen

as equally good. The first-order kinetics model (4) is only

able to describe a straight line in semi-logarithmic scale

(Fig. 3, gray dashed line). It is unadaptable and offers low

R2 (0.47–0.90). Because of very low goodness of fit, the

meaning of its parameters is not relevant.

Table 4 gives the optimized values of the parameters of

the mathematical models as a function of treatment time. In

the Weibull model (9), the values of n and b decrease with

a higher applied E-field. In the logistic model (13), the

value of the parameter x decreases with higher E. The

values of parameter s are very similar. The values of rincrease with longer pulses, as expected (higher value,

steeper slope). In the adapted Gompertz model (15), the

values of parameter A decrease, which means a lower

asymptote is reached. The values of parameters B0 and B1

also decrease, which means faster cell death with longer

pulses applied. In the Geeraerd model (18), Nres corre-

sponds to the remaining surviving cells and decreases with

higher E (similar to parameter A in the Gompertz model).

Parameter k corresponds to the speed of decrease and also

increases, both as expected.


as a Function of Electric Field

In Fig. 4 (8, 100 ls pulses, 1 Hz), it can be observed that

all models look very similar. The difference is in their

behavior at high E-fields, i.e., in extrapolation of the data.

In terms of goodness of fit, all four models on Fig. 4

(Peleg-Fermi (6), Weibull (10), logistic (14) and Gompertz

(16)) can be considered equal.

In Fig. 5 (90, 100 ls pulses, 1 Hz), however, the dif-

ferences among the models are more pronounced. They no

longer overlap as shown in Fig. 4. The Peleg-Fermi (6) and

Weibull models (10) go close but not exactly through the

Table 4 Calculated optimal values of parameters of mathematical models as a function of treatment time and R2 value for different electric field

values

Mathematical models Parameters Optimized values of parameters and R2 value

For 2.0 kV/cm (400 V) For 3.0 kV/cm (600 V) For 4.0 kV/cm (800 V)

First-order kinetics model (4) k 0.0008805 0.001382 0.001582

R2 0.9096 0.8237 0.4747

Weibull model (9) b 908 112 3.178

n 0.8915 0.5490 0.3105

R2 0.9135 0.9442 0.9225

Logistic model (13) x –3.925 –6.026 –20.760

r –5.059 –4.240 –2.815

s 3.659 3.532 4.078

R2 0.9346 0.9607 0.9260

Adapted Gompertz model (15) A –8.18 –14.15 –16.05

B0 1.419 0.3732 1.8e-6

B1 –0.0004339 –0.0004419 –0.001291

R2 0.9389 0.9661 0.8539

Geeraerd model (18) Nres 0.0005619 4.157e-5 4.014e-5

k 0.001004 0.002011 0.007824

R2 0.9390 0.9788 0.8411

In all the experiments, pulses of 100 ls duration with pulse repetition frequency 1 Hz were applied


123


68

experimental points (R2 between 0.90 and 0.91), since they

cannot model the lower asymptote but only a shoulder and

then a constant decrease of cell survival (on a semi-loga-

rithmic scale). The logistic and the adapted Gompertz

models, on the other hand, go exactly through the experi-

mental points (R2[ 0.99) and also look very similar. From

the point of view of the adaptability of the models, the

logistic and Gompertz models are better than the Weibull

(10) and Peleg-Fermi models (6).

Table 5 shows the results of fitting mathematical models

as a function of E-field for different pulse lengths. The

results of fitting the Peleg-Fermi model (6), the Weibull

model (10), the logistic model (14), and the adapted

Gompertz model (16) as a function of E-field are presented.

For each model, the optimal values of the parameters and

R2 value for three different pulse durations (50, 100 and

200 ls) are shown.

Table 6 presents the results of fitting mathematical

models as a function of E-field for different numbers of

pulses. Optimal values of parameters and R2 values for

each fit are shown. The pulse length was held fixed at

100 ls. It can be seen that R2 values are relatively high for

Table 5 Calculated optimal

values of parameters of

mathematical models as a

function of electric field and R2

for different lengths of the

pulses

Mathematical models Parameters Optimal values of parameters and R2 value

For 50 ls For 100 ls For 200 ls

Peleg-Fermi model (6) Ec/kV/cm 2.766 2.344 2.001

k/kV/cm 0.4160 0.2677 0.2871

R2 0.9968 0.9975 0.9836

Weibull model (10) b 2.992 2.408 1.936

n 2.831 3.645 2.695

R2 0.9900 0.9964 0.9615

Logistic model (14) x –1.555 –3.780 –2.961

r –0.963 –1.888 –2.082

s 3.383 3.537 2.916

R2 0.9966 0.9951 0.9809

Adapted Gompertz model (16) A –5.348 –19.080 –7.692

B0 3.528 2.683 4.149

B1 –1.013 –0.6438 –1.505

R2 0.9987 0.9991 0.9961

In all the experiments, 8 pulses with pulse repetition frequency 1 Hz were applied

Table 6 Optimized values of parameters of mathematical models as a function of E and R2 for different numbers of the pulses; the column 8

pulses is the same as the column 100 ls in Table 5

Mathematical models Parameters and R2 Optimal values of parameters and R2 value

For 8 pulses For 30 pulses For 50 pulses For 70 pulses For 90 pulses

Peleg-Fermi model (6) Ec (kV/cm) 2.3440 0.9720 0.9517 0.4298 0.4853

k (kV/cm) 0.2677 0.4260 0.2446 0.2833 0.2852

R2 0.9975 0.8959 0.9637 0.9085 0.9148

Weibull model (10) b 2.336 1.030 0.731 0.388 0.416

n 3.410 1.439 1.480 1.076 1.098

R2 0.9989 0.9652 0.9408 0.9004 0.9039

Logistic model (14) x –3.78 –2.93 –4.99 –4.58 –4.63

r –1.888 –1.474 –3.308 –5.029 –4.483

s 3.537 2.260 2.243 1.754 1.863

R2 0.9951 0.9918 0.9989 0.9825 0.9995

Adapted Gompertz model (16) A –19.08 –7.23 –11.74 –10.82 –10.96

B0 2.683 2.439 3.849 3.372 3.348

B1 –0.643 –1.221 –1.894 –2.221 –2.037

R2 0.9991 0.9987 0.9999 0.9865 0.9999

In all the experiments, pulses of 100 ls duration with pulse repetition frequency 1 Hz were applied


123


69

all the fits in Tables 5 and 6. The trends of the models are

therefore analyzed more carefully for each of the models

separately in the following paragraphs.

The plotted optimized Peleg-Fermi model (6) is pre-

sented in Fig. 6a. The Peleg-Fermi model (6) has an

additional two models, which describe Ec0 (7) and k (8) as

functions of the number of pulses (N). Optimal parameters

of these two models (7), (8) are plotted in Fig. 6b. In

Fig. 6a, it can be seen that for higher numbers of pulses (50

or more), the model does not go exactly through the

experimental points. For 8 and for 30 pulses, the Peleg-

Fermi model describes the data very well, since there are

no problems with modeling the lower asymptote. In

Fig. 6b, it can be seen that Eq. (7) fits the experimental

points well (black circles). However, Eq. (8) does not fit

the data well (white squares on Fig. 6b), R2 = 0.049.

Equation (8) suggests an exponential dependence of k on

the number of pulses, while in our data the value of k is

almost constant.

Table 7 presents additional results of fitting the Peleg-

Fermi model, in which optimal values of the parameters

and R2 value for each fit of Eqs. (7) and (8) are shown. As

already observed in Fig. 6b, Eq. (8) does not describe our

data well. It can be observed that the goodness of fit is

relatively high for the Peleg-Fermi model ([0.89) for dif-

ferent durations (Table 4), as well for different numbers of

pulses (Table 5).

The Weibull model as a function of E-field (10) has a

similar shape as the Peleg-Fermi model (6) (Fig. 5). The

Weibull model cannot describe a sigmoid shape in semi-

logarithmic scale, so a deviation at higher E-field of more

than 8 pulses is noticeable (compare gray solid lines in

Figs. 4 and 5). The meaning of the parameters of the

Weibull model has not yet been established (10) and there

is also no trend in the value of parameter n in our results.

The value of parameter b decreases with longer pulses

(Table 5) and with a higher number of pulses applied

(Table 6).

The logistic (14) model is more adaptable and has a

concave (Fig. 4) or sigmoid shape (Fig. 5) in semi-loga-

rithmic scale. When fitting the logistic model (14) to the

results of different pulse lengths (Table 5), r decreases

(faster death). Parameter s denotes where on the x axis the

decrease is fastest. It is similar for all pulse lengths

(Table 5). When only 8 pulses of different lengths are

applied (Table 5), the asymptote is not reached (Fig. 1a).

Although the model predicts an asymptote, it is outside the

Table 7 Calculated optimal values of parameters of additional Peleg-

Fermi mathematical models (7), (8) for Ec and k as functions of N

Mathematical models Parameters Optimal values

of parameters

and R2 value

Peleg-Fermi mathematical

model for Ec(N) (7)

Eco (kV/cm) 2.734

k1 0.02506

R2 0.9237

Peleg-Fermi mathematical

model for k(N) (8)

k0 (kV/cm) 0.3259

k2 0.001598

R2 0.04916

0 2000 4000 6000 8000 1000010−6

10−5

10−4

10−3

10−2

10−1

100

NxT /µs

S/-

ExperimentsFirst order kinetics model (4)Weibull model (9)Logistic model (13)Gompertz model (15)Geeraerd model (18)

First order kinetics

Weibull

Logistic

Gompertz

Geeraerd

R^2=0.944

R^2=0.966

R^2=0.961

R^2=0.824

R^2=0.979

Fig. 3 Mathematical models (lines) and experimental results (sym-

bols) showing cell survival as a function of the treatment time (3 kV/

cm, 100 ls, 1 Hz). On y axis, there is the proportion of the surviving

cells (S) in logarithmic scale. For each fit, R2 value is shown. Except

for the first-order kinetics model (4), all the models offer a good fit

(R2[ 0.94). We found the adapted Gompertz (15) and the Geeraerd

model (18) to be the most suitable

0 1 2 3 410−6

10−5

10−4

10−3

10−2

10−1

100

E / kV/cm

S/-

ExperimentsPeleg−Fermi model (6)Weibull model (10)Logistic model (14)Gompertz model (16)

Peleg-Fermi

Weibull

Logistic

Gompertz

R^2=0.996

R^2=0.998

R^2=0.995

R^2=0.999

8 pulses

5


bols) showing cell survival as a function of electric field (8 pulses,

100 ls, 1 Hz). On y axis, there is the proportion of the surviving cells

(S) in logarithmic scale. For each fit, R2 value is shown. All the

models as a function of electric field offer a similarly good fit

(R2[ 0.99). We found the adapted Gompertz (16), the Peleg-Fermi

model (6)–(8), and maybe the logistic (14) model to be the most

suitable


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70

range in which our models are valid. In this case, the value

of x is not relevant since the models are not meant for

extrapolation of the data. When fitting the logistic model to

the results of different numbers of pulses, the parameters

cannot be so easily explained. Parameter x is similar with

more pulses applied (Table 6) since we reach a similar

lower asymptote (Fig. 1b). Parameter r is similar for 70

and 90 pulses and on average higher than for 30 and 50

pulses. This means faster cell death when 70 or 90 pulses

are applied. Parameter s mostly decreases with a higher

number of pulses applied (Table 6), which means that cells

die at lower E-fields when more pulses are applied.

When applying the adapted Gompertz model (16) to the

results of different pulse lengths, the values of parameter

A are quite different (Table 5) because the lower asymptote

was not reached in the experiments (Fig. 2a). Parameter

A also does not have a trend with different numbers of

pulses applied (Table 6), but the reason could be that with

50, 70, or 90 pulses, there is a similar proportion of sur-

viving cells at higher E-field (around 10-5). Parameters B0

and B1 in Table 5 have similar values since the experi-

mental values for different lengths of the applied pulses are

similar. The value of B0 is similar for different numbers of

pulses (Table 6) since it denotes the length of the upper

asymptote, which is similar for all lengths of pulses

(Fig. 5). B1 decreases with more pulses (Table 6), which

means faster death with more pulses applied.

Discussion

Several mathematical models are able to describe experi-

mental results. The most appropriate models as a function

of treatment time are the adapted Gompertz (15) and the

Geeraerd models (18). The logistic model (13) can be used

but a clearer meaning of its parameters needs to be estab-

lished. The most appropriate models as a function of

electric field (E-field) were the Peleg-Fermi (6), the logistic

(14), and the adapted Gompertz models (16). Mathematical

models of cell survival could thus be integrated into

treatment planning of electrochemotherapy and irreversible

0 1 2 3 4 510−6

10−5

10−4

10−3

10−2

10−1

100

E/ kV/cm

ExperimentsPeleg−Fermi model (6)Weibull model (10)Logistic model (14)Gompertz model (16)

Peleg-Fermi

Weibull

Logistic

Gompertz

R^2=0.904

R^2=0.999

R^2=0.915

R^2=0.999

S/-

90 pulses


bols) showing cell survival as a function of electric field (90 pulses,

100 ls, 1 Hz). On y axis, there is the proportion of the surviving cells

in logarithmic scale. For each fit, R2 value is shown. Experimental

values at 3 kV/cm and at 4 kV/cm (the lower asymptote) are on the

limit of our detection. We found the adapted Gompertz (16), the

Peleg-Fermi model (6)–(8), and maybe the logistic (14) model to be

the most suitable

0 1 2 3 4 510−6

10−5

10−4

10−3

10−2

10−1

100

S/-

E / kV/cm

8 pulses30 pulses50 pulses70 pulses90 pulses

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

N

kV/cm

Eck

a b

R^2=0.924

R^2=0.049

R^2=0.998R^2=0.896

R^2=0.909

R^2=0.915

R^2=0.964

Fig. 6 The Peleg-Fermi model (6) for different numbers of pulses

applied. For each fit, R2 value is shown. a Symbols are the

experimental values for 1 Hz, 100 ls, lines show the Peleg-Fermi

model (6). b Symbols are the optimized values of Ec and k, lines show

the optimized mathematical models (7) and (8). On y axis, there are

the values of Ec and k in kV/cm. Since the Peleg-Fermi model (6)

incorporates dependence on E as well as on N (7), (8) it already

connects two treatment parameters (electric field and number of the

pulses) and can therefore be used more easily than other models

investigated in this study


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71

electroporation as a method of tissue ablation. It must be

emphasized that the models should not be extrapolated,

since they predict different behaviors at very high E-fields

or very long treatment time. Some of them keep on

decreasing and some of them reach a stable value on a

semi-logarithmic scale.

Experimental Considerations

The electrical parameters chosen for the experiments were

similar to electrochemotherapy and irreversible electropo-

ration electrical parameters typically used in vivo. For

electrochemotherapy parameters, we used a fixed number

of pulses (8) and we varied the length of the pulses. In

electrochemotherapy treatments, 100 ls pulses are usually

used but to evaluate the trend of the parameters of the

mathematical models we also applied 50 ls and 200 lspulses (results on Fig. 2a). For irreversible electroporation

parameters, we tried to cover the parameter space as

equally as possible. In an orthogonal space, where on one

axis there was number of the pulses (N) and on the other E-

field, we equidistantly sampled it by increasing the voltage

by 200 V and the number of pulses by 30 (results on

Fig. 2b).

An important aspect of irreversible electroporation

experiments is the effect of increased temperature. Because

of high voltage, many pulses, and high current, the tem-

perature in the tissue or (as in our case) in the cell sus-

pension can be increased considerably by Joule heating

(Zupanic and Miklavcic 2011). Heating can change the

conductivity of the cells, as well as damage them (Neal

et al. 2012). Cell death could therefore be a thermal and not

electrical effect (Garcia et al. 2014). The temperature of the

suspension was therefore measured before and within 5 s

after the end of application of 90 pulses, 4.0 kV/cm (the

most severe electrical parameters employed in our study).

Even with the most severe electrical parameters, the tem-

perature within 5 s after the end of pulse application did

not surpass 40 �C. This proved that, under our experi-

mental conditions, cell death can indeed be considered

solely as a consequence of irreversible electroporation.

The percentage of surviving cells was first evaluated

using tetrazolium based assay (MTS assay). The MTS

assay, however, proved not suitable for distinguishing

between low proportions of surviving cells and it cannot be

used to quantify the number of living cells exactly. The

MTS assay is based on measurements of absorbance, which

is then correlated to the number of metabolically active

cells. After reaching 2 % of the surviving/metabolically

active cells, the number did not drop, no matter how much

higher an E-field or how many more pulses we applied.

Since irreversible electroporation can be used successfully

to treat tumors, a lower percentage of surviving cells

should be achievable. In addition, metabolic activity and

the ability to divide do not necessarily correlate. We

therefore decided to use clonogenic assay, which requires

more time than the MTS assay but enables exact quantifi-

cation of the number of clonogenic cells. With more severe

treatments, we could detect as low as 1 surviving cell in

25,000 (4 9 10-5 survival). In some experiments, the

survival was lower than 4 9 10-5 because the final pro-

portion of surviving cells was calculated as a mean over at

least four repetitions. Often no cells survived (0 survival)

with severe treatments. In calculating the mean, the 0

survival caused the final proportion of the survival to be

lower than 4 9 10-5. The detection limit of our clonogenic

assay was thus reached at approximately 4 9 10-5. The

lower asymptote that can be observed in Fig. 2b for 50, 70,

and 90 pulses at 3.0 and 4.0 kV/cm could be a consequence

of the detection limit. For more precise (and lower), pro-

portions of surviving cells at high E-field values and many

pulses, more cells should be seeded, which can be achieved

using a denser cell suspension. However, more precise

results with denser cell suspension are perhaps not even

needed. In vivo, the last few clonogenic cells seem to be

eradicated by the immune system when performing elec-

trochemotherapy (Calvet et al. 2014; Mir et al. 1992; Sersa

et al. 1997), as well as irreversible electroporation (Neal

et al. 2013). The proportion of cells needed to kill to cause

a complete response and destroy the whole tumor should be

determined in future studies.

In our experimental results, we determined that the

transition area between maximum and minimum survival

gets narrower, i.e., the death of cells is quicker with higher

numbers of pulses applied (Fig. 1b). This is in agreement

with the theoretical predictions made (Garcia et al. 2014)

using the Peleg-Fermi model. The authors predicted that

the transition zone between electroporated and non-elec-

troporated tissue becomes sharper when more pulses are

applied.

There are also other parameters of the electric pulses,

and biological parameters, which could affect the survival

of cells. For example, the pulse repetition frequency in our

experiments was always 1 Hz and when calculating the

treatment time as t ¼ NxT ; we ignored the effect of pulse

repetition frequency. There are contradicting studies that

report on its effect on cell survival (Pakhomova et al. 2013;

Pucihar et al. 2002; Silve et al. 2014). The effect of pulse

repetition frequency on the shape of survival curves needs

also to be investigated in future studies. It must be

emphasized that even if t1 ¼ t2 ¼ N1T1 ¼ N2T2 his cannot

be necessarily understood as equal if T1 6¼ T2 and N1 6¼ N2:

Different repetition frequencies affect cell permeabiliza-

tion, and cell survival and temperature increase differently.

We therefore present and discuss the results of different

numbers and different lengths of applied pulses separately.


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72

The value of E-field applied in the tissue is needed for

correct prediction of surviving cells. At the moment, the

most reliable method of determining E-field in tissues is

numerical modeling. However, in future, the E-field in

tissue could be monitored using current density imaging

and magnetic resonance electrical impedance tomography

(Kranjc et al. 2012, 2015). Cell survival in tissue could be

correlated even better by taking into account conductivity

changes (Kranjc et al. 2014) measured during application

of the pulses.

One possible problem with prediction of cell death in

tissues is the use of mathematical models fitted in vitro in

an in vivo environment. Tissues, unlike cell suspensions,

are heterogeneous; there are connections between cells;

cells are irregularly shaped; extracellular fluid is more

conductive than the pulsing buffer used in our study; and

there is an immune system present. Before starting clinical

studies, the parameters of electric pulses are first tested on

cell lines. In the past, good correlation was found between

the behavior of cells in vitro and in vivo. We expect sur-

vival curves in tissues to have a similar shape as in our

in vitro study. The question is whether there will be a lower

asymptote present or survival as a function of treatment

time or E-field will keep decreasing. Our models can

describe both options. The optimal values of the parameters

depend on the sensitivity of the cells to the electric pulses

and will probably be different. If the threshold values of the

E-field for reversible and irreversible electroporation for

different tissues are compared, different values can be

found. For example, in vivo the threshold values of the E-

field for muscle (8 9 100 ls pulses) for reversible elec-

troporation have been determined to be 0.08 kV/cm and

0.2 kV/cm (parallel and perpendicular directions, respec-

tively) and for irreversible electroporation to be 0.4 kV/cm

(the same for parallel and perpendicular directions of

muscle fibers) (Corovic et al. 2010, 2012). In vivo the

threshold for irreversible electroporation of healthy pros-

tate tissue has been determined to be 1 kV/cm (90 9 70 lspulses) (Neal et al. 2014), and for healthy brain tissue

0.5 kV/cm (90 9 50 ls pulses) (Garcia et al. 2010). For

healthy liver tissue, the threshold for reversible electropo-

ration has been reported to be 0.36 kV/cm and for irre-

versible electroporation 0.64 kV/cm (Miklavcic et al.

2000). The thresholds thus seem to be different for dif-

ferent tissues (Jiang et al. 2015). In vitro the thresholds are

usually higher and different for different cell lines: for

reversible electroporation around 0.4 kV/cm and for irre-

versible electroporation around 1.0 kV/cm for 8 9 100 lspulses (Cemazar et al. 1998). The curves in vivo can

therefore be expected to have a similar shape but they will

be scaled according to the thresholds for different types of

tissue.

Mathematical Modeling

It was mentioned in the Introduction section that predictive

power is one of the three most important criteria for

choosing the model (in addition to goodness of fit and

trends of values of the optimized parameters of the mod-

els). In our current study, however, predictive power was

not assessed. For assessing predictive power, our optimized

models must be validated on the samples for which the

survival will be predicted. In our case, the models will be

used in predicting death of tissues in electrochemotherapy

and irreversible electroporation. Validation of the models

on tissues is beyond the scope of this paper but must be

done before implementing the models in actual treatment

planning of electroporation-based treatments. The reader

should also note that all the models approach 0 survival

asymptotically but can never actually reach 0 survival. As

already discussed, it is still not known how many cells must

be killed to achieve a complete response of the tumor.

Based on the fact that the immune system seems to erad-

icate the last remaining tumor cells, it seems that our

models adequately describe 0 to 100 % survival. It remains

to be established, however, what percentage of cells actu-

ally needs to be killed by irreversible electroporation.


as a Function of Treatment Time

We fitted the first-order kinetics (4), the Weibull (9), the

logistic (13), the adapted Gompertz (15), and the Geeraerd

(18) models to the experimental data as a function of

treatment time. At 1.0 kV/cm (200 V), the percentage of

surviving cells decreased to 68 % for 90 pulses, 100 ls,1 Hz. The results of fitting the models to 200 V are thus not

presented, since the decrease in survival was too small to

be relevant for describing cell death due to electroporation.

It can be seen in Fig. 3 that, except for the first-order

kinetics model (4), all the models describe the experi-

mental points well. From the point of goodness of fit, the

Weibull (9), the logistic (13), the adapted Gompertz (15),

and the Geeraerd model (18) are equal. The next criterion

is the trend of parameters which is to be discussed for each

model separately in the next paragraph.

The first-order kinetics model (4) has a very low R2

(Table 4) and it is not suitable for describing cell death. It

is still very often used for describing microbial inactiva-

tion. (Peleg 2006) stated that the first-order kinetics model

is popular because any data can be described with it if the

data are sampled too sparsely. The second reason for its

popularity is its long history. In the Weibull model (9), the

parameters have a trend. However, in many studies, it has

been shown that parameter n is not connected to any


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73

biological or other parameter (Alvarez et al. 2003; Mafart

et al. 2002; Stone et al. 2009; van Boekel 2002). It only

describes the shape of the curve (concave, convex, linear).

The Weibull model is often used because it is highly

adaptable and can describe different shapes. Because the

Weibull model was used in many previous studies, but the

meaning of the parameters was not established in any of

them, the Weibull model is most likely not suitable for

predicting cell death after electroporation. The logistic

model (14) has a high R2 and most of its parameters can be

connected to some biological parameter. It may be suitable

for predicting cell death due to electroporation but the

meaning of its parameters must be more clearly defined. In

the adapted Gompertz model (16), both parameters B0 and

B1 behave as expected (shorter upper asymptote and stee-

per decline). The Geeraerd model was defined for the

shapes of curves just like ours—first the number of the

cells exponentially decreases and, after a certain treatment

time, it reaches a lower asymptote. The R2 value was high

and there was a trend of the values of the parameters.

It can be concluded that, as a function of treatment time,

adapted Gompertz and Geeraerd models are suitable, while

the logistic model has potential but should be tested with

more electrical parameters.


as a Function of Electric Field

It can be seen in Fig. 4 that if there is no lower asymptote

present, all the models describe the data well and have a

similar R2 value. In Fig. 5, a lower asymptote is present and

the goodness of fit is different for different models. Looking

at Figs. 4 and 5, it can be said that the logistic (14) and the

adapted Gompertz models (16) are most suitable. In the

Weibull model (10) (Tables 5, 6), there is no trend in the

values of the optimized parameters. The Weibull model is

not suitable for the use in treatment planning. The adapted

Gompertz model (16) has high R2 and the values of its

parameters can be explained. The adapted Gompertz model

is thus suitable for describing cell death after electroporation.

The logistic model (14) is highly adaptable. Because of

the detection limit, not all values of the parameters behave

as expected. Until the detection limit is reached (Fig. 2a),

all the parameters can be explained (Table 5). It can

therefore be said that the logistic model is probably suitable

but it should be tested on a larger dataset.

We mentioned before that the Peleg-Fermi model (6)

does not well describe cell death for higher numbers of the

pulses. The reason is that it is not suitable for describing

lower asymptotes. However, it is very likely that the lower

asymptote is a consequence of the detection limit of the

clonogenic assay. In this case, it is not problematic that the

lower asymptote cannot be described. If it is discovered

in vivo that there is a lower asymptote present, the use-

fulness of the Peleg-Fermi model will have to be evaluated

separately for in vivo data. When the Peleg-Fermi model

(6) was fitted to our in vitro experimental results, the E-

field was the independent variable and the number of the

pulses or their length was the parameters. Unfortunately,

with different lengths of applied pulses, we could not

model the change of k(N) and Ec(N), since there is no

model to connect k and Ec with the length of the applied

pulses. With different numbers of applied pulses (Table 6),

we could also evaluate models for Ec(N) (7) and k(N) (8)

(Table 7). The value of Ec decreases with a higher number

of pulses (Table 6), is in agreement with our understanding

of Ec as a critical E-field (Pucihar et al. 2011) and can be

described using the proposed model (7). Ec changes less

with a higher number of pulses. An even higher number of

pulses applied would probably not lower the critical elec-

tric field but most likely only increase the heating. Values

of k are approximately similar for all different numbers of

applied pulses and they cannot be described using the

proposed model (8). One reason may be the sensitivity of

the clonogenic assay, as mentioned before. With more

pulses, there could be even lower proportions of surviving

cells, the model would be steeper and the value of

parameter k would decrease. In (Golberg and Rubinsky

2010), the model was fitted to experimental data of up to 10

pulses applied, while in our study we fitted it to up to 90

pulses applied. Equation (8) may be exponential for up to

10 pulses applied but for more pulses it seems more like a

constant. Equation (8) should be verified on tissues to see

whether there is an exponential dependency. We assume

that the Peleg-Fermi model (6), (7) will be suitable for use

in treatment planning of electrochemotherapy and irre-

versible electroporation, while the dependency of param-

eter k on the number of pulses (8) remains questionable.

We next compared the values of our optimized param-

eters to the values reported in the literature. Our values of

Ec and k are lower than in (Golberg and Rubinsky 2010)

and optimized to describe the experimental results for 8–90

pulses. The authors in (Golberg and Rubinsky 2010) opti-

mized their parameters to 1–10 pulses, whereby the

decrease of the number of cells in dependency on the E-

field is slower and smaller than for more pulses. This could

explain the lower values of k and Ec as well as the non-

exponential dependence of Eq. (8). The Peleg-Fermi model

(6) seems the most promising of all the models analyzed in

this study, since it also includes dependency on the number

of pulses.

It can be concluded that the Peleg-Fermi (6), the adapted

Gompertz (16), and probably also the logistic model (14)

can all be used for describing cell death due to electropo-

ration and could all be used in treatment planning of

electrochemotherapy and irreversible electroporation.


123


74

Since the Peleg-Fermi model incorporates dependence on

E-field as well as on the number of pulses it already con-

nects two electrical parameters and can therefore be used

more easily, while such connections still have to be

determined for other models.

Conclusion

Mathematical models can describe cell death after elec-

troporation. Hopefully, they can also be used in treatment

planning of electrochemotherapy and irreversible electro-

poration as a method of tissue ablation. Mathematical

models suitable for treatment planning have to describe the

data well, have predictive power, and their parameters have

to have a trend. Using the probability of cell death, the

treatment plan can be made more reliable and also more

comprehensive. Instead of displaying the electric field

around the electrodes, the probability of tissue destruction

around the electrodes can be shown. In this study, it was

shown that not only the Peleg-Fermi but also other models

are suitable for describing in vitro experimental results.

However, it still needs to be determined whether the pro-

posed mathematical models also have predictive and not

just descriptive power. Our results are valid only for one

cell line in suspension under our experimental conditions.

The question is whether mathematical models could also be

translated to tissues and more complex geometries. This

validation should be done on tissues since the models will

be used on tissues. Applying mathematical models of sur-

vival to tissues is thus the next step.

Acknowledgments This study was supported by the Slovenian

Research Agency (ARRS) and conducted within the scope of the

Electroporation in Biology and Medicine European Associated Lab-

oratory (LEA-EBAM). Experimental work was performed in the

infrastructure center ‘Cellular Electrical Engineering’ IP-0510. The

authors would like to thank Dr. Bor Kos for his helpful comments

about fitting of the models and numerical modeling and Lea Vuka-

novic for her help with the experiments in the laboratory.

Appendix: Numerical Model of TemperatureDistribution

We calculated temperature distribution to determine that 90,

100 ls pulses of 4 kV/cm and 1 Hz repetition frequency do

not induce significant Joule heating. A numerical model of a

drop of cell suspension between parallel plate electrodes was

made in ComsolMultiphysics (v4.4, Comsol, Sweden) using

Electric Currents, Heat Transfer, and Multiphysics modules

in time-dependent analysis. Electrodes were modeled as two

blocks of 20 9 10 9 1 mm and a drop of cell suspension

was modeled as a block of 7 9 7 9 2 mm. Two boundaries,

one on each electrode, were modeled as terminals, with

?400 V and-400 V in the first 90 s of the simulation, while

the other boundaries were electrically insulated. Similar as in

(Garcia et al. 2011a, b), only one pulse was applied for 90 s,

but we multiplied the Joule heating by the duty cycle (du-

ration/period) to adjust the amount of delivered energy. We

ran the simulation for an additional 5 s after the pulse

application to validate the model with our temperature

measurements, since measurements of temperature were

made within 5 s after the pulse application.

The change of conductivity due to cell electroporation

was disregarded in the model, since our cell suspension

was dilute. The values of parameters used in the simulation

are shown in Table 8. The properties of the cell suspension

(except for electrical conductivity, which is characteristic

of our electroporation buffer) are the same as for water.

The model was validated with current and temperature

measurements at 90, 100 ls pulses, 4 kV/cm, 1 Hz repe-

tition frequency. The predicted current (3.3 A) was in the

same range as the measured current (from 2.9 to 3.5 A).

Temperature measured within 5 s after the pulse applica-

tion was 37.0 �C, while the predicted temperature 5 s after

Table 8 Parameters, used in our numerical model, with their symbols, values, and units

Name of the parameter Symbol Value Units

Electrical conductivity rs 0.162[S/m] 9 (1 ? 0.02 9 (T[degC]-20)) S/m

re 1.73913[MS/m] 9 (1 ? 0.00094 9 (T[degC]-20))

Heat capacity at constant pressure Cps 4200 J/(kg K)

Cpe 500

Density qs 1000 kg/m3

qe 8000

Thermal conductivity ks 0.58 W/(m K)

ke 15

Relative permittivity es 80 –

ee 1

Subscript s denotes cell suspension and subscript e denotes electrodes


123


75

the pulse application was 37.6 �C. The model thus ade-

quately described our experiments.

The temperature distribution on the surface of the cell

suspension and on the electrodes, and a slice through the

drop of cell suspension after 90 pulses, is shown in Fig. 7.

Since the temperature of the cell suspension does not

exceed 42 �C, under our experimental conditions cell death

can indeed be ascribed to electroporation.

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Results and Discussion - Book chapter

Book chapter Title: Mathematical Models Describing Cell Death Due to Electroporation


Publication: Handbook of Electroporation

Publisher: Springer, Cham

Year: 2017

DOI: 10.1007/978-3-319-32886-7_13

Online ISBN: 978-3-319-32886-7

Print ISBN: 978-3-319-32885-0

79

Mathematical Models Describing Cell DeathDue to Electroporation

Janja Dermol and Damijan Miklavčič

ContentsIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Microbial Inactivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Electroporation-Based Medical Treatments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Models Used to Describe Cell Death in Suspension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6The First-Order Kinetics Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7The Hülsheger Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8The Peleg-Fermi Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9The Weibull Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11The Logistic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11The Adapted/Modified Gompertz Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12The Geeraerd Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12The Quadratic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12The Peleg-Penchina Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Models Used to Describe Cell Death In Vitro on Attached Cells and Cells in 3DTissue-Like Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Models of Cell Death Used in the Treatment Chamber and In Vivo . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

AbstractVarious models have been developed to describe microbial inactivation by pulsedelectric field treatment, and they have just recently been used for describingeukaryotic cell death due to irreversible electroporation. In microbial inactivation,the mathematical models of cell death enable the adaptation of the pulse param-eters to achieve sufficient microbial reduction at the lowest energy input while

J. Dermol (*) • D. MiklavčičFaculty of Electrical Engineering, University of Ljubljana, Ljubljana, Sloveniae-mail: [email protected]; [email protected]

# Springer International Publishing AG 2016D. Miklavcic, Handbook of Electroporation,DOI 10.1007/978-3-319-26779-1_13-1

1

Results and Discussion - Book Chapter

80



preserving flavor and sensitive compounds in the food. For precise prediction, thegeometry of the treatment chamber, the fluid flow, the temperature, and theelectric field distribution should also be taken into account. In electroporation-based medical treatments, currently, a deterministic critical value of electric fieldis used to delineate between the destroyed and the unaffected tissue. Conse-quently, tumor cells which have higher electroporation threshold than the exper-imentally determined may remain viable and cause incomplete tumor elimination.On the contrary, the more sensitive surrounding tissue could be damaged. Math-ematical models of cell death help to achieve sufficient cell death while minimiz-ing the damage to the surrounding vital structures. In this chapter, differentmodels are described which were already used for describing microbial inactiva-tion in liquid foods or eukaryotic cell death (the first order, the Hülsheger, thePeleg-Fermi, the Weibull, the logistic, the Adapted/Modified Gompertz, theGeeraerd, the quadratic, the Peleg-Penchina model). The cell death modelshave already been used for predicting survival in realistic setups like the treatmentchamber in microbial inactivation and different electrode geometries and tissuesin irreversible electroporation treatments. In conclusion, cell death models areuseful in predicting the treatment outcome. Unfortunately, since the mechanismsof cell death due to electroporation are not yet fully elucidated, the models areempiric. There is no direct connection between the parameters of the models andthe biological/electrical parameters. Thus, it is unclear which model is the mostappropriate to use. The models have to be optimized for each specific cell typeand electric pulses separately. The transferability from the in vitro to the in vivolevel is questionable.

KeywordsMicrobial inactivation • Predictive models • Numerical modeling • Treatmentplanning • Food pasteurization

Introduction

Mathematical modeling is becoming indispensable in the field of life sciences. Itenables description and prediction of a response of a biological system knowing theexcitation and other parameters, and gives insights into the mechanisms of phenom-ena. The number of biological experiments can be decreased which decreases thetime and costs needed to obtain results. In the field of electroporation, models existon different levels – molecules, lipid bilayers, cells, and tissues. Cell death orinactivation models are used for the description of microbial inactivation andrecently also eukaryotic cell death in irreversible electroporation as nonthermalsoft tissue ablation. In this chapter, the statistical models of cell death are describedon the level of cells and tissues.

Currently, there is no complete explanation of cell death due to electroporation(“▶Cell Death Due to Electroporation”), neither for prokaryotic nor eukaryotic

2 J. Dermol and D. Miklavčič


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http://link.springer.com/Cell Death Due to Electroporation

cells. The cell death is believed to be caused by the following mechanisms. First,high electric pulses can cause membrane destruction and necrotic cell death. Second,there is the apoptotic cell death, reasons for which are not yet clear. Third, in vivo,last of the clonogenic cells are destroyed by the immune system (“▶ ImmunologicalResponse During Electroporation”). Thus, the currently used cell death models areempirical and were developed on the basis of the best fit to the experimental results.

Microbial Inactivation

In liquid food pasteurization by pulsed electric fields (PEF), short high-voltageelectric pulses are applied to decrease the number of microbes and quality-degradingenzymes, and extending the shelf-life while retaining flavor and nutrients (Lelieveldet al. 2007; Sun 2014). In the literature on cell death models, the expressionsbacterial and microbial cell death are both used. The microbe is a short expressionfor microorganisms not visible with a naked eye which includes besides bacteria alsoprotozoa, fungi, and viruses. Thus, bacteria are a subgroup of microorganisms. Sincecell death models have been used for describing cell death of various microorgan-isms (i.e., bacteria, yeast), expression microbial cell death models is used throughthis chapter. Mathematical modeling of microbial cell death (“▶Modeling MicrobialInactivation by Pulsed Electric Field”) is necessary to adapt the pulse parameters andachieve sufficient decrease of microorganisms at a minimum energy input whilepreserving the sensitive compounds in the food (Huang et al. 2012). In the foodindustry, at least five log reductions of bacteria are required for food pasteurization.Applying too gentle pulses may cause food spoilage and health-related problemswhile applying too severe pulses may cause Joule heating and decrease the quality offood. The cell death models have been used on a variety of liquid food (e.g., fruitjuices, milk, water, liquid egg yolks) or models of such liquids. When predicting themicrobial inactivation by the PEF treatment, the geometry of the treatment chamberis modeled, and the electric field, the temperature distribution, and fluid flow are(numerically) calculated (Gerlach et al. 2008). Microbial cell death prediction can beincluded using the cell death models as a function of the electric field, treatment time,the number of the applied pulses, and/or the pulse repetition frequency (Huang et al.2013). The inactivation of the quality-degrading enzymes and the degradation of thehealth-related compounds can also be modeled with similar models as the inactiva-tion of microorganisms due to their similar kinetics.

Electroporation-Based Medical Treatments

When treating eukaryotic cells with electroporation, currently, the three main appli-cations are the electrochemotherapy, the irreversible electroporation (“▶TissueAblation by Irreversible Electroporation”), and the gene electrotransfer for genetherapy and DNA vaccination (Yarmush et al. 2014). When treating tumors withelectrochemotherapy, pulses of standard parameters and fixed electrode

Mathematical Models Describing Cell Death Due to Electroporation 3


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http://link.springer.com/Modeling Microbial Inactivation by Pulsed Electric Field


http://link.springer.com/Tissue Ablation by Irreversible Electroporation


configurations can be used. However, if the tumor is larger than 2 cm and/or has anirregular shape, a variable electrode configuration should be used. For the irrevers-ible electroporation treatments (Jiang et al. 2015), protocols are already set up in thecommercially available electroporators, but the ablated volumes do not alwayscorrelate to the predicted ones (Bhutiani et al. 2016). In the electrochemotherapy,irreversible electroporation, and also in gene electrotransfer treatments, the need forpatient-specific treatment planning (“▶Treatment Planning for Electroche-motherapy and Irreversible Electroporation of Deep-Seated Tumors”) is on therise. The number, the geometry of the electrodes, and the parameters of the electricpulses must be optimized considering the specifics of the cause and the treatment(Županič et al. 2012). In electrochemotherapy, all tumor cells must be reversiblypermeabilized, while the surrounding tissue, especially critical structures, should beundamaged. In irreversible electroporation, the tumor must be irreversiblyelectroporated without significant thermal damage, while the surrounding tissue,especially critical structures like nerves and vessels, should remain largelyundamaged. In gene electrotransfer, cells must be reversibly permeabilized, andtheir viability must remain high. Currently, an experimentally determined criticalelectric field is used to predict and delineate between the alive and the irreversiblyelectroporated tissue – the response is modeled by a step function (Fig. 1). Theassumption of a step function is, however, too simplistic even for cell suspensions ofthe same cell line (Canatella et al. 2001; Dermol and Miklavčič 2015) since cellsamong other differ in size and position in a cell cycle. In tissues, especially in tumors,this inhomogeneity is amplified since cells additionally differ in shape and tissues arecomprised of different cell types. In a tumor, in addition to the tumor cells, stromalcells are present in the microenvironment. Therefore, in reality, the transition fromalive to irreversibly electroporated state is continuous and distributed over a range ofelectric field values. The width of this range, as well as the threshold voltage, alsodepend on the number and length of the applied pulses (Canatella et al. 2001; Garciaet al. 2014).

When predicting the survival with the fixed threshold, the cells which have highercritical threshold than the experimentally determined can survive and cause incom-plete tumor elimination or its later recurrence. On the other hand, applying moresevere electric pulses than necessary can cause tissue necrosis, excessive Jouleheating, and thermal damage. By using cell death models to predict the treatedvolume, the efficiency of the electroporation-based medical treatments and therapiescan be increased. Cell death models allow interpolation and predict survival atparameters which were not experimentally determined. The last few of theclonogenic cells are eradicated by the immune system (“▶ Immune ResponseAfter Electroporation and Electrochemotherapy”) (Yarmush et al. 2014) whichshould be considered when determining the tolerated percentage of the survivedcells in electroporation-based treatments. A complete eradication of all tumor cellsmay not be needed.

In electrochemotherapy and gene electrotransfer, using cell death models is notenough to correctly predict the treatment outcome. In the electrochemotherapytreatments, the application of electric pulses increases the permeability of the cell



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membrane to the chemotherapeutics. Two commonly used chemotherapeutics arecisplatin and bleomycin. Cell death is mostly caused by the cytotoxic effects ofchemotherapeutics, although, in the vicinity of the electrodes, cells can be irrevers-ibly electroporated. To correctly predict the affected regions, a model of permeabilityto cytotoxic drugs and their transport across the membrane as a function of electricpulses should be included in the treatment planning of electrochemotherapy (Dermoland Miklavčič 2014). Namely, when the number of the molecules of chemothera-peutic entering each cell can be calculated, it can be predicted whether the cell willdie or not. In the gene electrotherapy, electric pulses increase the permeability of thecell membrane to the DNA. The exact mechanisms of gene electrotherapy arecomplex and also depend on genes. The presumed steps in the process are electroper-meabilization of the cell membrane, electrophoretic migration of the DNA towardsmembrane, DNA/membrane interaction, DNA translocation across the membrane,intracellular migration of DNA through the cytoplasm, DNA passage through thenuclear envelope, and gene expression (Rosazza et al. 2016). To correctly predictwhere the transfection will occur, each step should be modeled.

Fig. 1 The comparison of the currently used and the proposed way of modeling cell death. The redline shows the currently used step response where below a critical electric field all cells are regardedalive and above all dead. The gray and dark lines show the suggested cell death models whichpredict a gradual decrease in cell survival. Black circles are the experimentally determined values. Itis clear that the experimental values are better described with different cell death models than with astep response



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Models Used to Describe Cell Death in Suspension

Ideally, cell death model would include all the parameters, important for cell death.A problem arises since the exact mechanisms of cell death due to electroporation arenot completely known. Cell death depends on many parameters, among others on theelectric field; the duration and the number of the applied pulses; the pulse repetitionfrequency; the properties of the cells; their size, shape, orientation in the electricfield; their concentration in the treated medium; the properties of the treated mediumor food; pH; temperature; configuration of the electrodes or the treatment chamber;and the concentration of the chemotherapeutics (in electrochemotherapy). The mostimportant parameters for cell death should be identified and included in the model. Itshould also be determined which parameters can be excluded to simplify the modelsand the fitting. In summary, the cell death model due to electroporation shoulddescribe the experimental results well, include all the important electrical andbiological parameters, have high predictive power, and a minimum number ofparameters to be optimized.

Various models have been developed and used to describe microbial cell deathdue to pulsed electric field (PEF) treatment in liquid food pasteurization (Álvarezet al. 2003; Peleg 2006; San Martín et al. 2007; Huang et al. 2012). Only recently,interest to develop and use such models has been expressed in tissue ablation due toirreversible electroporation, i.e., IRE treatment. Most of the models used, however,are empirical, and their parameters have no physical meaning and even have noparameters that could be explicitly linked to treatment parameters such as electricfield, pulse duration, the number of pulses, and pulse shape. There is also no solidlink demonstrated between cell membrane permeabilization and cell death.

In microbial cell death models, the independent variable is usually the treatmenttime which is easily determined in the case of thermal, irradiation, high pressure,ultrasound, or other continuous treatment. The treatment time of the PEF treatment ismore difficult to determine since multiple short pulses are applied at different pulserepetition frequencies. Nevertheless, the treatment time (t) is usually calculated bymultiplying the number of pulses with the duration of one pulse (t ¼ NT ), whereN denotes the number of the pulses and T the duration of one pulse. This way ofcalculating the treatment time assumes that only the product of N and T affects thesurvival but not the duration or the number of the pulses by itself. The effect of thepulse repetition frequency is also neglected. In microbial cell death models, theelectric field, the number of pulses, or their repetition frequency were used asindependent variables. With the exception of the quadratic model, however, nomodel so far included more than two independent variables.

The electric field is the most important parameter in electroporation-based treat-ments (Miklavčič et al. 2006). The survival curves as a function of the electric fieldare usually composed of three parts: a shoulder (upper asymptote) which is thenfollowed by a steep decline in cell number and eventually, a tail (lower asymptote) isreached. The tail can be a caused by either a resistant subpopulation of cells or thereached limit of the survival assay. Several different curves can describe the survival



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data. When presenting the survival curves and the cell death models, they should beplotted on a semilogarithmic scale. There, deviations and small percentages ofsurvival are more easily noticed. The quality of the fit should be evaluated usingvarious statistical measures: the goodness-of-fit (R2), the root-mean-square error(RMSE), and/or accuracy parameter (Af). R

2 is a statistical measure of how closethe regression line is to the experimental data; RMSE is a measure of the averagedeviation between the observed and predicted data; and Af measures the accuracy ofthe estimates obtained by the models. Other criteria for evaluating the suitability of amodel are the number of the models’ parameters (which should be low), theinclusion of more independent variables, and a high predictive power. In continua-tion, models which were already used for describing microbial or eukaryotic celldeath are described.

The First-Order Kinetics Model

The first-order kinetics model was developed almost a century ago (Huang et al.2012). It derives from the assumption that all cells in a population are equallysensitive to the treatment. When used in PEF treatments, it describes cell death asa function of the treatment time:

S tð Þ ¼ exp �ktð Þ; (1)

where S denotes the survival, t the treatment time, and k is a first-order parameter,i.e., the speed of the decrease. The model predicts that the viability decreasesimmediately – there is no shoulder at short treatment time. The first-order kineticmodel did not provide a good fit to electroporation treatment of eukaryotic cells(Dermol and Miklavčič 2015). It has also been observed that it does not describe allmicrobial inactivation data well. It was stated that the model describes the datasufficiently well only when the data is sampled too scarcely (Peleg 2006). Thedeviations from the first-order kinetics model could among others be explained bythe statistical distribution of cell radii (Lebovka and Vorobiev 2004).

The first-order parameter k was found to be temperature dependent which couldbe modeled in thermal treatments as well as in PEF treatments. In PEF treatments,the temperature of the sample increases due to the Joule heating and consequentlythe electrical conductivity of the sample is also increased. The parameter k could bemodeled by Arrhenius equation as:

k ¼ ktexp � EA

RT

� �(2)

Where kt is the rate constant at the reference temperature, EA the activation energy,R is the gas constant, and T is the sample temperature in Kelvins.

Traditionally, the sensitivity of microorganisms to static treatments is describedusing the decimal reduction time (Dt). It denotes the treatment time needed to obtain



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one decimal reduction of the cell population, i.e., 10 % of the initial populationsurvives. It is defined as Dt ¼ 2:303

k since the first-order kinetics model is based onnatural and the decimal reduction time on the decimal logarithm.

If the treated cell population consists of two populations of which each has itsfirst-order dynamics, the biphasic model can be used:

S tð Þ ¼ f e�k1t þ 1� fð Þe�k2t; (3)

where f denotes the proportion of the first subpopulation in the whole population,and k1 and k2 are the first order parameters of the first and the second subpopulation,respectively. In case there are more subpopulations, the decrease of each subpopu-lation can be modeled with its own exponential factor which is then added to the(Eq. 3).

The Hülsheger Model

Hülsheger studied the effect of electric field on the inactivation of E. coli (Huanget al. 2012). He derived an empirical formula which has two independent variables –the treatment time (t) and the electric field (E). As a function of the electric field, thesurvival showed a linear decline when the electric field exceeded a certain criticalelectric field (Ec) which was modeled as:

S Eð Þ ¼ exp �bE E� Ecð Þð Þ; (4)

where S is the survival, bE is the regression coefficient, and Ec is the critical value ofelectric field below which there will be no inactivation (i.e., the lowest electric fieldthat causes inactivation). When E < Ec, the model predicts the survival above onewhich is incorrect and the survival must be fixed at one.

As a function of the treatment time, the results were modeled as:

S tð Þ ¼ exp �btlnt

tc

� �� ; (5)

where S is the survival, bt is the regression coefficient, and tc is the extrapolatedcritical value of treatment time below which there will be no inactivation (i.e., theshortest treatment time which causes inactivation). For treatment times shorter thanthe critical treatment time, the model predicts the survival to be more than 1. Thus,also here, for t < tc the survival should be fixed at one.

Both models Eqs. 4 and 5 were joined in:

S t,Eð Þ ¼ t

tc

� ��E�Eck

(6)



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where k is the independent inactivation constant, and other parameters have the samemeaning as in Eqs. 4 and 5. The parameters tc, Ec, and k were proposed to bemicroorganism dependent within a certain range of experimental parameters.

Although the Hülsheger model includes two independent variables, it cannotuniversally describe the experimentally determined cell death. With the Hülshegermodel, some microbial cell death results were possible to model, and some were not(Huang et al. 2012), while only one study unsuccessfully fit it to the eukaryotic celldeath data (Dermol and Miklavčič 2015).

The Peleg-Fermi Model

The Peleg-Fermi model has been successfully fit to various microbial inactivationdata (Huang et al. 2012) and was also the first to be used for describing eukaryoticcell death due to electroporation (Golberg and Rubinsky 2010). It derives from theFermi’s equation which is used to describe the behavior of materials at their glasstransition temperature. The Peleg-Fermi model is one of the most promising since itdescribes the data well and includes two independent variables – electric field (E)and the number of the applied pulses (N ):

S E,Nð Þ ¼ 1

1þ expE� Ec Nð Þ

k Nð Þ� � (7)

Ec Nð Þ ¼ Ec0exp �k1Nð Þ (8)

k Nð Þ ¼ k0exp �k2Nð Þ (9)

where Ec(N ) is the critical electric field where the survival drops to 50 %, k(N ) is thekinetic constant describing the slope of the curve, Ec0 is the intersection of Ec(N )with the y-axis, k0 (in the same units as the electric field), k1 and k2 are constantswhich change depending on the parameters of the pulses and the properties of thecells. The critical electric field depends on the number and length of the appliedpulses (Pucihar et al. 2011). For values below the critical electric field, the Peleg-Fermi model predicts survival to be 100 %. The Peleg-Fermi model describedvarious experimental data well, although the k(N ) and Ec(N ) did not always changeexponentially as a function of the pulse number (Dermol and Miklavčič 2015;Sharabi et al. 2016). Unfortunately, the pulse length and the pulse repetition fre-quency are not included in the model, and the dependency on them is includedindirectly via the parameters of the model. An example of fitting the Peleg-Fermimodel to the eukaryotic cell death due to irreversible electroporation in vitro isshown in Fig. 2.



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Fig. 2 The Peleg-Fermi model, fit to eukaryotic cell death data in a wide range of pulse number andelectric field values. (a) symbols show the experimental values for 100 μs, 1 Hz repetition frequencyat different pulse numbers (8, 30, 50, 70, and 90), lines show the fitted Peleg-Fermi model (Eq. 7).(b) symbols are the optimized values of Ec and k for different pulse numbers, lines show the fittedmodels for Ec(N ) and k(N ) (Eqs. 8 and 9) (Reprinted from Journal of Membrane Biology, Vol248/Issue 5, Dermol J, MiklavčičD, Mathematical Models Describing Chinese Hamster Ovary CellDeath Due to Electroporation In Vitro, Pages 865–881, Copyright 2015 Springer Science + Busi-ness Media New York)



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The Weibull Model

The Weibull model was originally used to describe the time to failure of electronicdevices after stress was imposed on them. A parallel can be made when stress, forexample, thermal, high pressure, or PEF treatment, is applied to biological cells.Time to cell death after the stress can be described using the Weibull model (SanMartín et al. 2007). The Weibull model was successfully used to describe microbial(Huang et al. 2012) and eukaryotic cell death (Dermol and Miklavčič 2015) usingvarious pulse parameters. Although the Weibull model is very adaptable, no clearconnection between the electrical/biological parameters and the optimized parame-ters of the model was made. In some cases, the scale parameter b was exponentiallydependent on the electric field. The model is:

S xð Þ ¼ exp � x

b

� �n� �; (10)

where x denotes the treatment time or the electric field, b is the scale (in the sameunits as x), and n is the shape parameter. The scale parameter determines thecharacteristic time or the characteristic electric field at which 63 % of the cells die.With different values of the shape parameter, the shape of the survival curve variesbetween the convex (n < 1), linear (n = 1), and concave curve (n > 1) in semi-logarithmic scale.

The Logistic Model

The logistic model can be used for describing distributions with a sharp peak andlong tails (Cole et al. 1993). When taking into account that survival of cells beforethe treatment is 100 %, the logistic model can be written as either Eq. 11 or 12,depending on what is chosen as the independent variable. The equations are:

S Eð Þ ¼ 10^ω

1þ exp 4σ τ � Eð Þω�1ð Þ� �

(11)

S tð Þ ¼ 10^ω

1þ exp 4σ τ � log10tð Þω�1ð Þ� �

(12)

where E denotes the electric field, t the treatment time, ω the common logarithm ofthe lower asymptote, σ the maximum slope, and τ the position of the maximumslope. Survival curves show the cumulative cell death as a function of the indepen-dent variable, i.e., applied electric field or treatment time. Thus, each data pointincludes cells which died due to the corresponding or less severe parameter. The celldeath distribution is obtained as a derivative of the cumulative distribution. Only celldeath due to the corresponding parameter is plotted. In electroporation treatments,the distribution of cell death with a sharp peak and long tails is obtained when the



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independent variable is either the electric field or the common logarithm of thetreatment time. The logistic model was successfully fit to inactivation of microor-ganisms by PEF treatment (Huang et al. 2012) and to eukaryotic cell death (Dermoland Miklavčič 2015).

The Adapted/Modified Gompertz Model

The Gompertz model is usually used for describing the growth of a tumor, but inmodified form, it can also describe cell death. Originally, it was used for describingbacterial (Linton et al. 1995) cell death due to thermal treatment, but later it wasextended to describing bacterial cell death due to pulsed electric field treatment andeukaryotic (Dermol and Miklavčič 2015) cell death due to electroporation. Themodel is written as:

S xð Þ ¼ exp A exp �e B0þB1xð Þ� �

� A exp �eB0� ��

(13)

where x denotes either the treatment time or the electric field, A is the naturallogarithm of the survival in the stationary phase (the natural logarithm of the tail),B0 is the length of the shoulder, and B1 the speed of the increase (when it is positive)or decrease (when it is negative) in cell number.

The Geeraerd Model

The Geeraerd model (Geeraerd et al. 2000) describes the exponential decrease ofcells including a tail which models the resistant cell subpopulation. Assuming theinitial survival is 100 %, the Geeraerd model can be written as:

S tð Þ ¼ 1� Nresð Þexp �ktð Þ þ Nres (14)

Where Nres represents the tail and k is the inactivation rate. Since the model does notalso include a shoulder, it cannot be used universally for all cell death results. TheGeeraerd model was so far successfully fit to experimental data of microbialinactivation after mild heat treatment and eukaryotic cell death due toelectroporation.

The Quadratic Model

The quadratic model is currently the only model which can model dependency ontwo or more independent variables. As the independent variable x, various param-eters have been used – the electric field, the treatment time, the pulse repetition



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frequency, the pH, and the concentration of some compound. The model is writtenas:

S xð Þ ¼ k1 þ k2xþ k3x2 (15)

where the parameter k1 is the central point of the system, k2 is the coefficient of thelinear effect, and k3 of the quadratic effect of the independent variable. It is alsopossible to fit more parameters at once by joining several quadratic models into one:

S x1, x2ð Þ ¼ k1 þ k2x1 þ k3x12 þ k4x2 þ k5x2

2 þ k6x1x2 (16)

where x1 and x2 denote two independent variables which were marked as the x in the(Eq. 15), k1 is the central point of the system, k2 and k4 represent the linear, k3 and k5the quadratic, and k6 the interactive effects of the independent variables. If some ofthe terms are nonsignificant under certain experimental conditions, they can beomitted from the model. By quadratic model fitting, the interaction between differentexperimental parameters can be determined. The quadratic model was successfullyfit to microbial inactivation results as well as to the inactivation of the enzymes andhealth-related compounds like vitamins (Huang et al. 2012).

The Peleg-Penchina Model

The Peleg-Penchina model is also an empirical model but can describe only theconvex curves (Peleg and Penchina 2000; Álvarez et al. 2003) in the semilogarith-mic scale. It is written as:

S tð Þ ¼ 10 ^ �m ln 1þ ktð Þð Þ (17)

where t is the treatment time of the PEF treatment; m and k are the parameters of themodel, which have to be optimized; and ln denotes the natural logarithm. Togetherwith the Eq. 17, the authors introduced a way of describing bacterial survival whenthe intensity of the lethal agent (e.g., temperature, chemical agent, PEF treatment)varies either between treatments or during one treatment.

As long as the mechanisms of cell death due to electroporation (PEF treatment)are not known, it is difficult to decide which model is superior to others, and they canall be regarded as equivalent. An example of fitting several cell death models toeukaryotic and bacterial cell death results is shown in Fig. 3.



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Fig. 3 (a) Different cell death models as a function of the electric field (the Peleg-Fermi, theWeibull, the Logistic, the Adapted Gompertz model) were fitted to the experimental data ofeukaryotic cell death when 90, 100 μs pulses were applied to the cell suspension. Symbols showthe experimental values and lines the models. It can be seen that several models describe theexperimental data well Observed values are marked by circles, model 1 by a bold line, model 2 by adotted line, model 3 by a thin line, and model 4 by a dashed line. (Reprinted from Journal ofMembrane Biology, Vol 248/Issue 5, Dermol J, Miklavčič D, Mathematical Models DescribingChinese Hamster Ovary Cell Death Due to Electroporation In Vitro, Pages 865–881, Copyright2015 Springer Science + Business Media New York 2015). (b) Different cell death models as afunction of the treatment time were fitted to the survival of E. coli after pulsed electric fieldtreatment. Model 1 is the biphasic model, model 2 is the sigmoidal equation, model 3 is the Weibullmodel, and model 4 is the Peleg-Penchina model. A good fit was obtained with all tested models(Reprinted from Innovative Food Science & Emerging Technologies, Vol 4/edition number4, Alvarez I, Virto R, Raso J, Condon S, Comparing predicting models for the Escherichia coliinactivation by pulsed electric fields, Pages No 195–202, Copyright (2003), with permission fromElsevier)



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Models Used to Describe Cell Death In Vitro on Attached Cellsand Cells in 3D Tissue-Like Structure

In liquid food pasteurization, the microorganisms are suspended in the food. How-ever, they can also grow in layers, for example, on surgical meshes. Surgical mesh isthin mesh which supports or reinforces damaged tissue. The cell death models wereapplied to PEF treatments of medical mesh implants, on which 2D layers of bacteriacan form as a biofilm (“▶Electroporation of Biofilms”) (Khan et al. 2016). Theauthors successfully described the data using the Weibull model.

The cell death models have not yet been applied to the attached or 3D eukaryoticcell models. The optimization of the models in vitro would need to be done anewsince the critical electric fields are different for the attached than for the suspendedcells due to a different shape, density, and connections between cells (Pucihar et al.2006; Towhidi et al. 2008).

Models of Cell Death Used in the Treatment Chamber and In Vivo

Microorganisms grow in suspensions and 2D layers. Thus, there are no reports onusing the mathematical models to describe PEF inactivation of microorganismsin vivo. There is, however, an example of modeling the treatment chamber (“▶Opti-mization of Pulsed Electric Field Treatment Chamber”) with included suspension ofbacteria. Huang et al. predicted the inactivation of bacteria in watermelon juice byfirst building a 2D numerical model of the treatment chamber and using theHülsheger model to describe the inactivation (Huang et al. 2013). Authors obtainedvery good predictions. Cells respond to the electric field to which they are exposed.In treatment chambers (as well as in tissue when performing the IRE treatment), theelectric field can be highly inhomogeneous. In future, to obtain the optimal condi-tions for microbial inactivation, first optimization of the treatment chambers’ geom-etry and the parameters of electric pulses should be done.

There are just a few reports of using cell death models to describe and predict theextent of eukaryotic cell death due to electroporation in vivo. Golberg and Rubinskywere first to suggest using statistical cell death models in vivo (Golberg andRubinsky 2010). They fit the Peleg-Fermi model to the results of an in vitro studyof electroporation of prostate cell line. Then, the fitted Peleg-Fermi model wastheoretically applied to the 2D case of tissue electroporation using needle electrodes.The spatial distribution of probability of cell death the authors obtained is shown inFig. 4. It was shown that there exists an area where the cell death probability rangesfrom 0 % to 100 % and the deterministic critical electric field is not an optimalchoice for predicting the electroporation-based medical treatment outcome.

A theoretical study of commercially available bipolar electrodes used in irrevers-ible electroporation treatments was done by Garcia et al. (2014). The authorsdetermined the extent of the cell death caused by the heating and by the electropo-ration when standard IRE pulses were applied. The thermal damage was evaluatedusing the Arrhenius integral and the electrical damage using the Peleg-Fermi model.



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http://link.springer.com/Optimization of Pulsed Electric Field Treatment Chamber


The electrical damage due to electroporation around the bipolar electrodes is shownin Fig. 5. Different combinations of pulse number and electric field lead to differentprobabilities of cell death as shown in Fig. 6.

A study in vivo was performed by Sharabi et al., who analyzed the brainelectroporation using the Peleg-Fermi model to describe cell death as well as thedisruption of the blood–brain barrier (Sharabi et al. 2016). The authors concludedthat the Peleg-Fermi model could be successfully used to describe the electropora-tion of rat brain, although they found that the Ec(N ) was better described using powerthan an exponential function, especially in the range of a high number of pulses(more than 90). As shown in (Eq. 8), Ec(N ) in the Peleg-Fermi model can beexpressed as Ec Nð Þ ¼ Ec0exp �k1Nð Þ . Sharabi et al. obtained better fit using theequation:

Ec Nð Þ ¼ Ec0N�k1 ; (18)

where Ec0 denotes the critical electric field, N the number of the applied pulses, andk1 is a constant.

In electroporation-based medical treatments, the desired use of the models is intissues in vivo. It is questionable to what extent the in vitro optimized models can beused in vivo. The thresholds for cell death due to electroporation in vivo (Jiang et al.2015) seem to be different than in vitro (Dermol and Miklavčič 2015). In vivo, the

Fig. 4 The Peleg-Fermi model was fit to the 2D model of prostate tissue treated by irreversibleelectroporation using needle electrodes. The legend on the right shows the probability of cell death.Around the electrodes exists an area where the probability of cell death is between 0 % and 100 %,thus the deterministic critical electric field is not sufficient for predicting cell death in vivo. (a–c)show the predicted cell death when a different number of 100 μs pulses of the same voltage (1.5arbitrary unit) are applied: (a) 10 pulses, (b) 50 pulses, (c) 100 pulses (Reprinted from BioMedicalEngineering OnLine, Vol 9, Golberg A, Rubinsky B, A statistical model for multidimensionalirreversible electroporation cell death in tissue, Page No 13, Copyright 2010 Golberg and Rubinsky;licensee BioMed Central Ltd.)



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cells are connected, they are of different size and shape, their density varies, there aredifferent types of cells in each tissue, and the immune system is present. It seems thatthe cell death models optimized in vitro may not be directly transferable in vivo butwould need to be optimized separately.

Fig. 5 The probability of cell death due to electroporation around the commercially availablebipolar electrodes at different pulse numbers (a – 30, b – 50, c – 70, and d – 90) when applying100 μs pulses of 3000 Vat 1 Hz. The transition zone where the cell death decreases from 100 % to0 % becomes sharper with increasing pulse number (Reprinted from PLoS ONE, Vol 9/Issue8, Garcia PA, Davalos RV, Miklavčič D, A Numerical Investigation of the Electric and ThermalCell Kill Distributions in Electroporation-Based Therapies in Tissue, Pages No e103083, Copyright2014 Garcia et al.)



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Conclusions

Using cell death models in treatment planning could increase the efficiency of PEFmicrobial inactivation and electroporation-based medical treatments. However, cur-rently, it is unclear which model is the most appropriate since they are all empirical,and mostly, there is no clear relation between the electrical/biological parameters andparameters of the cell death models. Although the cell death models describe datawell, they must be optimized for each cell type/tissue and different electric pulsesseparately. A direct translation of in vitro optimized models to an in vivo environ-ment is questionable. In future, cell death models should be based on mechanisms ofcell death due to electroporation, include all and only the relevant treatment param-eters, and describe and predict the cell death accurately.

Acknowledgments This study was supported by the Slovenian Research Agency (ARRS) andconducted within the scope of the Electroporation in Biology and Medicine European AssociatedLaboratory (LEA-EBAM).

Fig. 6 The probability of cell death due to electroporation as a function of pulse number andelectric field. The cell death is predicted by the Peleg-Fermi model optimized for the prostate cancercell death. The contours show the predicted cell death (denoted with numbers in bold) using variouscombinations of electric field and pulse number. To achieve 99.9 % cell death a certain minimalelectric field and a number of pulses should be applied (Reprinted from PLoS ONE, Vol 9/Issue8, Garcia PA, Davalos RV, MiklavčičD, A Numerical Investigation of the Electric and Thermal CellKill Distributions in Electroporation-Based Therapies in Tissue, Pages No e103083, Copyright2014 Garcia et al.)



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Cross-References

▶Cell Death Due to Electroporation▶Electroporation of Biofilms▶ Immune Response After Electroporation and Electrochemotherapy▶ Immunological Response During Electroporation▶Modeling Microbial Inactivation by Pulsed Electric Field▶Optimization of Pulsed Electric Field Treatment Chamber▶Tissue Ablation by Irreversible Electroporation▶Treatment Planning for Electrochemotherapy and Irreversible Electroporation ofDeep-Seated Tumors

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Paper 3 Title: Plasma membrane depolarization and permeabilization due to electric pulses in cell lines of

different excitability

Authors: Janja Dermol-Černe, Damijan Miklavčič, Matej Reberšek, Primož Mekuč, Sylvia. M. Bardet,

Ryan Burke, Delia Arnaud-Cormos, Philippe Leveque and Rodney O'Connor

Publication: Bioelectrochemistry, vol. 122, pp. 103-114, Aug. 2018.

DOI: 10.1016/j.bioelechem.2018.03.011


Quartile: Q1 (Biology)

Rank: 15/85 (Biology)

101

Bioelectrochemistry 122 (2018) 103–114

Contents lists available at ScienceDirect

Bioelectrochemistry

j ourna l homepage: www.e lsev ie r .com/ locate /b ioe lechem


Plasma membrane depolarization and permeabilization due to electricpulses in cell lines of different excitability

Janja Dermol-Černe a, Damijan Miklavčič a, Matej Reberšek a, PrimožMekuč a, Sylvia M. Bardet b, Ryan Burke b,Delia Arnaud-Cormos b, Philippe Leveque b, Rodney O'Connor c,⁎a University of Ljubljana, Faculty of Electrical Engineering, Tržaška cesta 25, SI-1000 Ljubljana, Sloveniab University of Limoges, CNRS, XLIM, UMR 7252, F-87000 Limoges, Francec École des Mines de Saint-Étienne, Department of Bioelectronics, Georges Charpak Campus, Centre Microélectronique de Provence, 880 Route de Mimet, 13120 Gardanne, France

Abbreviations: ECT, electrochemotherapy; IRE, irreverChinese hamster ovary cells; MEM, minimal essentialminimal eagle's medium; PMPI, plasma membrane poten⁎ Corresponding author.

E-mail addresses: [email protected] (J. [email protected] (D. Miklavčič), [email protected] (S.M. Bardet), delia.arnaud-c(D. Arnaud-Cormos), [email protected] (P. Leveq(R. O'Connor).

https://doi.org/10.1016/j.bioelechem.2018.03.0111567-5394/© 2018 Elsevier B.V. All rights reserved.

102

a b s t r a c t
a r t i c l e i n f o
Article history:Received 20 October 2017Received in revised form 13 February 2018Accepted 19 March 2018Available online 21 March 2018

In electroporation-basedmedical treatments, excitable tissues are treated, either intentionally (irreversible elec-troporation of brain cancer, gene electrotransfer or ablation of the heart muscle, gene electrotransfer of skeletalmuscles), or unintentionally (excitable tissues near the target area). We investigated how excitable and non-ex-citable cells respond to electric pulses, and if electroporation could be an effective treatment of the tumours of thecentral nervous system. For three non-excitable and one excitable cell line, we determined a strength-durationcurve for a single pulse of 10 ns–10 ms. The threshold for depolarization decreased with longer pulses and washigher for excitable cells. We modelled the response with the Lapicque curve and the Hodgkin-Huxley model.At 1 μs a plateau of excitabilitywas reachedwhich could explainwhyhigh-frequency irreversible electroporation(H-FIRE) electroporates but does not excite cells. We exposed cells to standard electrochemotherapy parameters(8 × 100 μs pulses, 1 Hz, different voltages). Cells behaved similarly which indicates that electroporation mostprobably occurs at the level of lipid bilayer, independently of the voltage-gated channels. These results couldbe used for optimization of electric pulses to achievemaximal permeabilization andminimal excitation/pain sen-sation. In the future, it should be establishedwhether the in vitro depolarization correlates to nerve/muscle stim-ulation and pain sensation in vivo.

© 2018 Elsevier B.V. All rights reserved.

Keywords:ElectroporationExcitable cellsYO-PRO-1®Strength-duration curveThe Hodgkin-Huxley modelLapicque curve

1. Introduction

Short, high-voltage pulses increase the permeability of cell mem-branes to different molecules (reversible electroporation) or cause celldeath (irreversible electroporation) [1–3]. Electroporation is used inbiotechnology, food-processing [4–6] and medicine [7], e.g. geneelectrotransfer [8–10], DNA vaccination [11–14], transdermal drug de-livery [15,16], IRE as a soft tissue ablation technique [17–20] andelectrochemotherapy [21–24].

In medical applications of electroporation, different types of tissuesand tumours are treated, among them, also excitable tissues, e.g. criticalstructures like nerves and spine. Non-excitable as well as excitable cells

sible electroporation; CHO cells,medium; DMEM, Dulbecco's

tial indicator; YP, YO-PRO-1®.

rmol-Černe),[email protected] (M. Reberšek),[email protected]), [email protected]

can be depolarized, i.e. their transmembrane potential increases. How-ever, only excitable cells can produce action potentials due to their ex-pression of a high density of voltage-gated channels which enableselectrical communication between cells [25]. Types of excitable cells in-clude neurones, muscle, endocrine and egg cells.

In the literature, there are several examples of electrochemotherapy,irreversible electroporation and gene electrotransfer of excitable tissuesby electric pulses. Brain cancer is treated with irreversible electropora-tion, and electric pulses can transiently disturb the blood-brain-barrierand allow chemotherapeutics to enter the brain [26–31]. Treating pros-tate cancer can affect the neurovascular bundle [32,33], treating bonemetastases can affect nerves [21,34], treating tumours in the spine canaffect the spinal cord [34]. When treating tumours in other parts ofthe body, electrodes will invariably be in the vicinity of the nerves ormuscles where the electric field is high enough for excitation or evenpermeabilization. Electric pulses are also used for ablation ofmyocardialtissue to treat atrial fibrillation [35–37]. Muscles are a popular target forgene electrotransfer as they are easily accessible and transfected[38,39]. Among them, the heart can be electroporated to treat ischemia[40,41]. Other examples of the application of electric pulses to excitablecells include electroporation of neurons as a labelling technique which

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enables subsequent analyses [42–46]. Studies exist on the influence ofnsPEF on intracellular calcium and release of catecholamine in endo-crine cells [47–49].

Several studies showed that the effect of electric pulses on the func-tionality of excitable tissueswas only short-term. After irreversible elec-troporation the nerves of different animal models recovered electro-physiologically, histologically and functionally [34,50–52] or at leastshowed a potential for regeneration [53]. After electroporation of indi-vidual rat neocortex neurons, in vitro and in vivo, themembrane poten-tial, the action potential waveform and passive membrane propertiesremained unchanged [54]. After pulmonary vein ablation with electro-poration, the histology and functionality of phrenic nerve remained un-changed [55]. There was no histological damage on nerves in theneurovascular bundle when treating prostate with irreversible electro-poration [51].

One of the main drawbacks to the treatment of tissues with pulsedelectric fields is the discomfort and pain associated with repeated elec-trical stimulation [56–59], the need to administer muscle relaxants andanaesthesia [60] and synchronization with the electrocardiogram [61–63]. The neurons responsible for pain sensation (nociceptors) can bestimulated by electric pulses [64,65]. An important advance of the treat-mentwould be to determine a point atwhichmaximumpermeability ofthemembrane could be achievedwhileminimizing excitation of the ex-posed excitable cells. In electrochemotherapy, for example, this wouldtranslate to maximumdrug delivery to tumour cells withminimum tis-sue damage to surrounding regions, reduced pain experienced by thepatient, and minimal use of muscle relaxants. One suggested option isapplying short bipolar pulses which do not cause muscle contractionbut come at the expense of delivering higher energy than with longermonopolar pulses [66].

Questions that need to be answered include, (i) whether excitableand non-excitable tissues respond similarly to electroporation pulses,(ii) can electroporation be an effective treatment of cancers of the cen-tral nervous system, (iii) are the properties of surrounding excitable tis-sues significantly altered or damaged due to the treatment. In our paper,we evaluated the depolarization and membrane permeability of fourcell lines, one excitable and three non-excitable. For each cell line, wedetermined the strength-duration curve to one pulse of durations be-tween 10 ns and 10 ms. Additionally, response of excitable cell linewas modelled with the Lapicque curve and the Hodgkin-Huxleymodel. Lapicque curve is an empirical description of the strength-dura-tion curve and the Hodgkin-Huxley model is a set of differential equa-tions, describing the dynamics of voltage-gated channels when theyare exposed to electric stimulus. Our study shows that higher electricfields are needed for depolarization of excitable cell line than for depo-larization of non-excitable cell lines. We compared the depolarizationresults with previously experimentally determined electroporation re-sults modelled with the Saulis model [67]. We explain the lack of exci-tation after electroporation with short 1 μs pulses as well as highfrequency short bipolar pulses (H-FIRE). This lack of excitability withshort bipolar pulses can be explained by reaching a plateau of excitationand electroporation around 1 μs. We also determined the permeabilitycurve to pulses of standard electrochemotherapy parameters (8 × 100μs, 1 Hz at different electric field amplitudes). All four cell lines werepermeabilized at approximately the same electric field, and the perme-abilization curveswere similarly shapedwhich indicates that electropo-ration is independent of voltage-gated channels.

2. Materials and methods

2.1. Cell culture/preparation

Four cell lines of different excitability were used (Fig. 1). CHO Chi-nese hamster ovary cells (European Collection of Authenticated CellCultures ECACC, CHO-K1, cat. no. 85051005, obtained directly fromthe repository), U-87 MG human glioblastoma cells (ECACC, Public

Health England, cat. no. 89081402), and HT22 immortalizedmouse hip-pocampal neurons (The Salk Institute, La Jolla, CA)were grown in anhu-midified environment at 37 °C at 5% CO2. CHO cells were grown in theHam-F12 (Sigma-Aldrich, Germany), U-87MG inMEM (Sigma-Aldrich,Germany) and HT22 cells in DMEM (Sigma-Aldrich, Germany) growthmedium. All growth media were supplemented with 10% fetal bovineserum, L-glutamine and antibiotics. Cells were grown either on Poly-Ly-sine (Sigma-Aldrich, Germany) coated 22 mm glass coverslips whichwere put inside a plastic ring or on 40 mm Petri dishes (TPP, Austria).The HT22 cell line was differentiated by 48 h incubation in theNeurobasal medium (Gibco, USA) supplemented with 0.5 mM L-gluta-mine and B-27 serum-free supplement as advised by the manufacturer.In this time, the cells stopped dividing and changed their morphology(Fig. 1c vs d). We tested the excitability by chemically exciting thecells by varying the concentration of extracellular potassium ions andcomparing the response to a 1 ms pulse at 0.6 kV/cm.

2.2. Cell labelling – depolarization and membrane permeability

In the depolarization experiments, we labelled the cells with theplasma membrane potential indicator fluorescent dye (PMPI) of theFLIPR Membrane Potential Assays Kit (Molecular Devices, USA). For30 min before the experiments, cells were incubated at 37 °C and 5%CO2 in Live Cell Imaging Solution (Fischer Scientific, France) supple-mented with 1 mg/ml 20% D-glucose (Gibco, France) and 0.5 μl/mlPMPI. PMPI consists of a two-part system which includes a fluorescentanionic voltage-sensor and a quencher. When the interior of the cellhas a relatively negative charge (in the state of resting potential or repo-larization), the anion dye remains bound to the external surface of theplasma membrane. In this state, the quencher in the extracellular fluidpreventsfluorescence excitation. During depolarization the voltage sen-sor translocates to the interior of the plasmamembrane. This transloca-tion is reversible as we observe cells return to their base-levelfluorescence approximately 30min after exposure to pulse. An increasein fluorescence is observed because the quenching agent is restricted tothe extracellular environment. The time-constant of sensor transloca-tion is in the range of seconds [68].

To determine whether cells were excitable, cells were exposed todifferent extracellular concentrations of potassium ions (2.5 mM,25 mM, 50 mM, 75 mM, 100 mM, 140 mM), while the NaCl (140 mMin the original Live Cell Imaging Solution) was substituted for KCl inan equimolar manner. The buffer was changed every 5 min, and thecells were continuously imaged. Increased concentration of extracellu-lar K+ ions increases the equilibrium potential of K+ and causes depo-larization. Chemical depolarization is slow and does not cause actionpotential but accommodation. In electrical depolarization experiments,images were acquired every 30 s for 15–30 min, and pulse was deliv-ered 5 min after the beginning of imaging.

In permeability experiments, we labelled the cells with the YO-PRO-1® (YP) (Molecular Probes, USA). Right before the experiments, thegrowth medium was changed with the Live Cell Imaging Solution sup-plemented with 1 mg/ml 20% D-glucose (Gibco, France) and 1 μM YP.Images were acquired every 3 s for 6 min, and 8 × 100 μs pulses weredelivered 30 s after the beginning of the imaging.

2.3. Exposure of cells to electric pulses

Three different pulse generatorswere used, each for a different pulseduration range. For 10 ns pulses, we used a commercially availablensPEF (nanosecond pulsed electric fields) generator (FPG 10-1NM-T,FID Technology, Germany) where the electric field was numerically de-termined [69] (Fig. 3a). For pulse exposures of 550 ns–1 μs, we used alaboratory prototype pulse generator (University of Ljubljana) basedon H-bridge digital amplifier with 1 kV MOSFETs (DE275-102N06A,IXYS, USA) [70] (Fig. 3b). For pulse exposures of 10 μs–10 ms we useda commercially available BetaTech electroporator (Electro cell B10,

103

Fig. 1. Phase-contrast images of all four cell lines used in experiments. a - CHO, b - U-87 MG, c - undifferentiated HT22 and d - differentiated HT22.

105J. Dermol-Černe et al. / Bioelectrochemistry 122 (2018) 103–114Results and Discussion - Paper 3

BetaTech, France). In all experiments, we measured the delivered volt-age and current by oscilloscope (DPO 4104, Tektronix, USA orWavesurfer 422, 200 MHz, LeCroy, USA), voltage probe (tap-off 245NMFFP-100, Barth Electronics Technology, USA for 10 ns pulses or dif-ferential probe ADP305, LeCroy, USA for longer pulses) and a currentprobe (CP030, LeCroy, USA or Pearson current monitor model 2877,Pearson Electronics, Inc., USA). We used different systems as not allpulse generators were available in both laboratories.

For 10 ns pulses, we used two stainless-steel needle electrodes witha 1.2mm gap (Fig. 2a) [69]. For pulses longer than 10 ns we used eitherstainless-steel wire electrodes with a 4 mm inter-electrode distance orPt/Ir wire electrodes with 1 mm, 2.2 mm or 5 mm inter-electrode dis-tance (Fig. 2b) [70], depending on the electric field needed and thepower limitations of the generators. The electric field in the middle be-tween the electrodes was nearly homogeneous and could be

Fig. 2. Photos of the electrodes, used in our experiments. a - electrodes used for delivering 10 ninter-electrode distance.

104

approximated as the applied voltage divided by the distance betweenthe electrodes [70].

In depolarization experiments, we delivered one pulse, and we var-ied the pulse duration (10 ns, 550 ns, 1 μs, 10 μs, 100 μs, 1 ms, 10 ms),and voltage to determine the minimum electric field intensity requiredto cause a change in membrane potential (depolarization) that was sta-tistically different from control conditions. This minimum electric fieldintensity was considered to be the depolarization threshold. In mem-brane permeability experiments, we delivered 8 × 100 μs pulses at1 Hz repetition frequency.

2.4. Fluorescence microscopy and measurement

Imaging was performed on two different fluorescent microscopysystems due to different availability of pulse generators in the two

s with 1.2 mm gap, b - electrodes used for delivering pulses longer than 10 ns with 5 mm

0 10 20 30 40 50 60

t/ns

0

2

4

6

8

10

Vk/

V

8.9 kV

5.3 kV

4.1 kV

2.6 kV

2.0 kV

0 0.4 0.8 1.2

0

100

200

300

400

V/V

t/ s

Fig. 3. Traces of pulses applied in the experiments. a - traces of 10 ns pulses generated by a commercial 10 ns nsPEF generator [69], b - a trace of 1 μs pulse generated by a prototypegenerator from University of Ljubljana [70]. Voltage V/kV or V/V is presented as a function of time t/ns or t/μs, a and b, respectively.


laboratories where the work was done. However, by repeating severalsamples on both systems, we determined that the difference in systemsdid not influence the results. Either a DMI6000 inverted microscope(Leica Microsystems, Germany) with EMCCD camera (EMCCD Evolve512, Roper, USA) or an AxioVert 200 inverted microscope (Zeiss, Ger-many) with VisiCam 1280 CCD camera (Visitron, Germany) and eithera 100× oil immersion or 40× dry objective for PMPI and 20× dry objec-tive for YP were used for experiments. Samples were excited with ap-propriate wavelengths using the Spectra 7 light engine (LumencorUSA) or a monochromator (High-Speed Polychromator, Visitron Sys-tems GmbH, Germany) and the emission lightwas selected through ap-propriate filters. Images were acquired using MetaFluor andMetaMorph PC software (both Molecular Devices, USA).

2.5. Image analysis

First, the backgroundwas subtracted, each cell was selected by usingfreehand tool, and its mean fluorescence was determined. In

depolarization experiments, we determined the maximal fluorescencein the first 2.5 min and normalized it to the base line (the value of fluo-rescence prior exposure to pulses). In permeability experiments, thecells do not take up any YP without the electric pulses applied. Thus,we reported the values either raw or normalized to the fluorescenceat maximal pulse parameters (8 × 100 μs, 1.2 kV/cm, 1 Hz repetitionfrequency).

2.6. Statistical analysis

The threshold of depolarization was determined using ANOVA tests.The threshold was determined as the lowest field intensity required toproduce a statistically significant membrane depolarization. The statis-tical parameters are given in the Appendix, Table A1. The threshold ofelectroporation (the fluorescence that was significantly different fromthe control) and the comparisons of the change inmaximalfluorescencebetween all four cell lines were determined using t-test (p b 0.05) inSigmaPlot v.11 (Systat Software, San Jose, CA).

3. Calculation/models

3.1. Cell excitability

Cell excitability models described the depolarization/action potential thresholds of the excitable differentiated HT22 cell line as the models weused are valid for excitable cells. We modelled the strength-duration curve with two models – the Lapicque curve [71] which is one of the mostoften used experimentally tested theoretical model and the Hodgkin-Huxley model [72] which is a phenomenological description of the activityof voltage-gated channels.

The Lapicque curve is in the form:

I ¼ b 1þ cT

� �; ð1Þ

where b denotes the rheobase, c the chronaxie and T the duration of the stimulus. Since we were controlling the electric field to which cells wereexposed, we substituted the current (I) with the electric field (E). For the rheobase value we took the depolarization threshold at applied 10 ms,i.e. 0.28 kV/cm. The chronaxie should thus be the pulse duration at twice the rheobase, i.e. around 6 μs. However, with these parameters, thecurve fits the data poorly, and thus we determined the parameter c to be 1.88 μs. A non-physiological value of parameter indicates that the Lapicquecurve is not an optimal choice for description of our data.

One of the classical models describing neuronal excitation is the Hodgkin-Huxley (HH) model. We numerically calculated the strength-duration curve via the HH model as described in [71]. For several pulse durations, we calculated the corresponding critical transcellular volt-age which triggers the action potential (Fig. 4). The K, Na, and leakage current were modelled separately for the anodic and cathodic pole ofthe cell and coupled via the equivalent circuit (Fig. 1A in [71]). Cells were modelled as idealized planar cells with two uniformly polarizedflat surfaces. The corresponding external electric field was calculated by dividing the transcellular voltage with the diameter of the cell. Wemodelled monopolar as well as bipolar pulses. The bipolar pulse consisted of a positive pulse immediately followed by a negative pulse. Theduration of the bipolar pulse is the duration of a separate positive or negative pulse– the whole duration was thus twice this value. The cellexcitability calculations were performed in Matlab, R2017a (Mathworks, USA). Shape of the strength-duration curve depends on the timeconstant of the membrane (Fig. 3 in [71]).

105

510150t/ms

-100

-80

-60

-40

-20

0

20

40

Vm/

V m

Action potential

Depolarization

Cathodal side of the cellAnodal side of the cell

Fig. 4. Example of depolarization without and with initiated action potential. Blue and orange lines denote the voltage at the cathodal and anodal pole of the cell, respectively. After thestimulus, only the orange line is visible as the blue and orange lines are one on top of another and overlap. In this example, one 1 ms pulse was applied just below and at the threshold.Induced transmembrane voltage Vm/mV is presented as a function of time t/ms. (For interpretation of the references to colour in this figure legend, the reader is referred to theweb versionof this article.)


3.2. Plasma membrane permeabilization

Permeabilization consisted of two sets of experiments – thosewith the application of one pulse (results previously published in [67]) and the ap-plication of 8 pulses (data acquired in this study).

When one pulsewas applied, twomodels were used – a time-dependent Schwann equation or theory of kinetics of pore formation. First, we usedthe time-dependent Schwann equation for prolate ellipsoidal cells [73,74]:

ΔVcrit ¼ ER1

2−R22

R1−R2

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR1

2−R22

q logR1 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR1

2−R22

q

R2

0

@

1

A

R2 cos φð ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR1

2 sin2φþ R22 cos2φ

q 1− exp −tτ

� �� ; ð2Þ

where the time constant of the membrane was 0.87 μs (τ) as determined in [67] and R1 = 27 μm± 7 μm and R2 = 10 μm± 5 μmwhich we deter-mined from131 differentiatedHT22 cells on phase-contrast images. The critical threshold of electroporation (Vcrit) 1.776 Vwas obtained by the least-square method, but the fit was poor (R2 = 0.53) Parameter t denotes time and φ the angle between the electric field and position on the plasmamembrane. Because we were interested in the maximal transmembrane voltage, we assumed φ = 0 (prolate ellipsoid oriented in the direction ofthe electric field. Electric field (E) was expressed and we calculated the critical applied electric field for electroporation.

Second, we used the expression for a fraction of electroporated cells as derived by [75].We used the optimized parameters for attached cells in amonolayer from (Eq. (5) and Fig. 5 in this [67]):

Fp E; tp� � ¼ 1− exp −kf Eð Þtp

� � ð3aÞ

0 5 10 15 20 25 30

t/min

0

1

2

3

4

5

- / )

f/f(

0

CHO

U-87 MG

HT22 undiff.

HT22 diff.

b

0 5 10 15 20 25 300

5

10

15

20

25

30

t/min

-/ )

f/f0

a

Fig. 5. Time-dynamics of electrical and chemical depolarization of all four cell lines. a) Chemical depolarization for all four cell lines. The K+ concentration was increased every 5 min. Thearrows mark which K+ concentration cells were exposed to from that time point further. b) Electrical depolarization when 1 pulse of 1 ms and 0.6 kV/cm was applied. In a and b theincrease over base level (Δf/f0)/arbitrary units is presented as a function of time t/min with shown 1 standard deviation.

106


kf Eð Þ ¼ 2πνR2

aexp −

ΔW0

kbT

� �Z π

−πexp πCmr2

εwεm

−1� �

2kbTK1ER cos θð Þ 1− exp

−tpK2τ

� �� þ Δφ

� �2

2

664

3

775dθ; ð3bÞ

where Fp denotes the fraction of electroporated cells (0.7), tp the duration of the pulse, ν the frequency of lateralfluctuations of the lipids (1011 s−1),Ris byfitting determined cell radius (15.9 μm) [67], a the area per lipidmolecule (0.6 nm2),ΔW0 the energy barrier for pore formation (46.4 kbT), kb theBoltzmann constant, T the temperature in kelvins, r the radius of the pore (0.32 nm), Cm the capacity of themembrane (1 μF/cm2), εw (78) and εm (4)the relative permittivity of the water in the pore and membrane, respectively, K1 (1.24) and K2 (2.56) two numerical parameters, τ the membranetime constant (0.87 μs), Δφ resting membrane voltage (−25 mV). The values were used as optimized by experimental data on attached CHO cellsand electroporation was determined by Fura-2AM [67].

When 8 pulses were applied, the normalized fluorescence curve was obtained by:

f n Eð Þ ¼ f Eð Þ− f E ¼ 0 kV=cmð Þf E ¼ 1:2

kVcm

� �− f E ¼ 0 kV=cmð Þ

ð4Þ

where fn(E) denotes the normalized fluorescence, f(E) the raw fluorescence and E the applied electric field. The normalized fluorescence curve wasthen fitted using a symmetric sigmoid to obtain permeabilization curve [76]:

p Eð Þ ¼ 1

1þ exp −E−E50%

b

� � ð5Þ

where p denotes the normalized permeabilization, E the applied electric field, E50% the electric field where 50% of the final fluorescence was reachedand b the width of the curve. Permeabilization curve is based on normalized fluorescence of the population and not the exact percentage of perme-abilized cells.

The rawvalues of YO-PRO-1-® fluorescencewere described using a first-order uptakemodel, which gave information on the resealing speed [77].The model was:

f tð Þ ¼ S 1− exp −tτ

� �� ð6Þ

where S is the reached plateau, t is the time, f is the raw fluorescence and τ is the time constant of the resealingwhen 63% of the permeable structuresin themembrane are resealed. A higher value of τmeans slower resealing than lower value of τ. For curve fitting, we usedMatlab R2015 (Mathworks,USA) and the Curve Fitting Toolbox. The goodness of the fit was evaluated using R-squared value, whose value closer to 1 indicates better fit.

10-8

10-6

10-4

10-2

T/s

10-4

10-3

10-2

10-1

100

)m

c/V

k(/

E :n

oita

zilib

ae

mr

eP

10 -5

10-4

10-3

10-2

10-1

100

)m

c/V

k(/

E :lait

net

op

noit

ca/

noit

azi

ral

op

eD

Model Saulis

Schwann's equation

Experiments Pucihar

Lapicque curve

Action potential - unipolar pulse

Action potential - bipolar pulse

Depolarization CHO

Depolarization U-87 MG

Depolarization HT22 undiff.

Depolarization HT22 diff.

Fig. 6. The depolarization strength-duration curve for CHO, U-87MG, undifferentiated anddifferentiated HT22 cells and permeabilization curve from [67] for CHO cells. Black linesand left y-axis show permeabilization to a single pulse of duration T/s. Grey lines andright y-axis show depolarization and action potential due to the application of a singlepulse of duration T/s. Depolarization was determined using the PMPI dye and theelectroporation by the Fura-2AM dye. Lines correspond to models and symbols toexperimental values. External electric field E/(kV/cm) is presented as a function of pulseduration T/s.

4. Results

Results are divided into two sections – cell depolarization andplasma membrane permeabilization after one pulse and plasma mem-brane permeabilization after pulses of standard electrochemotherapyparameters. Results in the cell depolarization section were obtainedwith the PMPI dye. Results in the plasma membrane permeabilizationsection were obtained with the YO-PRO-1® dye.

4.1. Cell depolarization and plasma membrane permeabilization after onepulse

Fig. 5 shows the response of excitable andnon-excitable cells to elec-trochemical and electrical depolarization. In Fig. 5a, the differential be-havior of cells to depolarization by increasing extracellular K+

concentrations is shown. In Fig. 5b, the results to electrical depolariza-tion after 1 pulse of 1ms (themost relevant point for neurostimulation)at 0.6 kV/cm is shown. We compared the intensity of the response andthus exposed all four cell lines to electric pulses of the same parameters.The increase in fluorescence was observed when the pulse was appliedat threshold or above-threshold electric fields. We also tested severaldifferent pulse durations and electric fields. Depolarization curves forother pulse durations were similar in shape and values as the curvesafter excitation with 1 ms pulse.

Fig. 6 shows the strength-duration depolarization curve (right y-axis) and strength-duration permeabilization curve (left y-axis). Bothaspects (depolarization and membrane permeabilization) were experi-mentally determined aswell asmodelled. The experimental depolariza-tion threshold was determined from the membrane depolarizationcurves, for each pulse duration and cell line separately. It is shown in

grey symbols – x denotes the CHO cells, Δ the U-87 MG cells,□ the un-differentiated HT22 cells and ○ the differentiated HT22 cells. The exactvalues are stated in Table 1. We can see that with increasing pulse

107

Table 1The depolarization thresholds for all tested pulse durations and cell lines. The results arethe same as in the Fig. 6. The asterisk (*) denotes that the threshold is only estimateddue to the variability of the data.

Electric field (kV/cm) 10 ns 550 ns 1 μs 10 μs 100 μs 1 ms 10 ms

CHO 44 2.0 1.2 0.45 0.30 0.15 0.10U-87 MG 34 2.2 1.4 0.60 0.35 0.20 0.12Undifferentiated HT22 34 2.0 1.4 0.70 0.50 0.35 0.24Differentiated HT22 52 2.0 1.7* 0.90 0.60 0.40 0.28

Table 2Parameters of the fitted symmetric sigmoid to the normalized data of YO-PRO-1® uptake.8 × 100 μs pulseswere delivered at repetition frequency1Hz. The numbers denote the op-timal value and the corresponding 95% confidence intervals.

Cell line E50% (kV/cm) b (kV/cm) R-squared

CHO 0.80 ± 0.06 0.15 ± 0.11 0.98U-87 MG 0.81 ± 0.10 0.13 ± 0.09 0.99HT22 undifferentiated 0.94 ± 0.07 0.12 ± 0.06 0.99HT22 differentiated 0.91 ± 0.07 0.12 ± 0.06 0.99


duration the depolarization threshold decreased. At most of the pulsedurations, the threshold for depolarization was the highest for the dif-ferentiated HT22 cell line. The statistical parameters of the analysis ofthe experimental depolarization threshold are shown in the Appendixin Table A1. The depolarization strength-duration curve was modelledusing the Lapicque curve and the Hodgkin-Huxley model. The modelsare only valid for excitable cell lineswhich have voltage-gated channels.In excitable cells we obtained excitability strength-duration curve andin non-excitable cell lineswe obtained depolarization strength-durationcurve. The modelled curves are presented with grey lines – in dash-dotline is the Lapicque curve, in solid line is the Hodgkin-Huxley model forthe unipolar pulses and in dashed line is the Hodgkin-Huxleymodel forbipolar pulses. The Hodgkin-Huxley model predicts a plateau at around1 μs and is also slightly better at describing the data than the Lapicquecurve.

Experimental electroporation thresholds are shown in filled blackcircles [67]. By the Saulis model modelled electroporation threshold ispresented in black lines. The dashed line is the Schwann's equation.Bothmodels follow a similar dynamics up to 1ms, but for longer pulses,the Saulismodel predicts lower electroporation thresholds andmatchesthe experimental data better. In the Schwann model, the threshold ofelectroporation was assumed to be constant for all pulse lengths, al-though it is possible that it changed. The Saulis model was obtainedby fitting the model to the experimental results [67] and thus followedthe experimental data better than the Schwann equation. Depolariza-tion and electroporation thresholds follow a similar dependency, al-though the electroporation thresholds are slightly higher than thedepolarization thresholds. The same dependency indicates that electro-poration and depolarization behave similarly as a function of the ap-plied electric field.

4.2. Plasma membrane permeability after pulses of standardelectrochemotherapy parameters

In plasma membrane permeability experiments, we delivered8 × 100 μs pulses of different voltage at 1 Hz repetition frequency.These parameters are the standard electrochemotherapy parametersand were used to evaluate the possibility of using electrochemotherapy

0 0.2 0.4 0.6 0.8 1.0 1.2

E/(kV/cm)

0

0.2

0.4

0.6

0.8

1

1.2

-/f n

CHO

U-87 MG

HT22 undiff.

HT22 diff.

a

Fig. 7.Normalized fluorescence (a) and permeabilization curve (b) of all four cell lines to YO-PR0.4 kV/cm (U-87 MG and CHO) or 0.6 kV/cm (undifferentiated and differentiated HT22 cells)sigmoid. Normalized fluorescence fn/− is presented as a function of applied electric field E/(kV

108

as treatment of excitable tissues or in the vicinity of excitable tissues.The normalized plasma membrane permeabilization curve (Eq. 4) toYO-PRO-1® is for all four cell lines shown in Fig. 7. The threshold of elec-troporation of the U-87 MG and CHO cells was 0.4 kV/cm while of theundifferentiated and differentiated HT22 cells it was 0.6 kV/cm. Thethreshold was determined as electric field intensity where the fluores-cence 6.5 min after pulse application was significantly higher than thefluorescence of the control. With the increase of the electric field, thepermeabilization also increased in a similar way for all four cell lines.The permeabilization curve could be described using a symmetric sig-moid (Eq. 5), which is shown in Fig. 7b. The symmetric sigmoid param-eters (Table 2) show that all four cell lines reached 50%permeabilization between 0.8 and 0.9 kV/cm (E50%).

An example of the time dynamics of the uptake at applied 8 × 100 μsat 1.2 kV/cm is shown in Fig. 8a. From the time-lapse of the YO-PRO-1®uptake, we obtained the maximal fluorescence. The maximal fluores-cence (Fig. 8b) was the highest for the U-87 MG cells and the lowestfor the CHO cells (CHO vs U-87 MG is just significant at p = 0.0468,CHO vs. undifferentiated HT22 p = 0.0002, CHO vs. differentiatedHT22p=0.0008.). Therewere no significant differences in themaximalfluorescence between the differentiated and undifferentiated HT22swhile for all other pairwise comparisons the difference was significant(t-test, p b 0.05). We obtained the resealing constant (τ) (Fig. 8c) byfitting a first-order uptake model (Eq. 6). The value of time constant τ(Fig. 8c) corresponds to the time when 63% of the pores in the mem-brane resealed. The resealingwas the fastest for the U-87MG cells, sim-ilar for the undifferentiated HT22s and CHO and slower for thedifferentiated HT22 cells. Statistical significance of the fit is obtainedby comparing the error bars which represent a statistical error of 5%.

5. Discussion

Our study we aimed at comparing the depolarization thresholds be-tween excitable and non-excitable cells, to determine if excitable andnon-excitable cells respond similarly to electroporation pulses, and todetermine if electroporation can be an effective treatment of cancersof the central nervous system. For each cell line, we determined thestrength-duration curve to one pulse of lengths between 10 ns and

0 0.2 0.4 0.6 0.8 1.0 1.2

E/(kV/cm)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-/f n

b

CHO

U-87 MG

HT22 undiff.

HT22 diff.

O-1®, 5 min after the pulse application. The threshold for electroporation was reached at. a) Normalized values, the bars are one standard deviation and b) the fitted symmetric/cm).

0 1 2 3 4 5 6 7

t/min

0

2

4

6

8

10

12

14

16

.u.

a/f

CHO

U-87 MG

a

CHO

U-8

7 M

G

HT22

undiff.

HT22

diff.

0

2

4

6

8

10

12

14

.u.

a/f

b

CHO

U-8

7 M

G

HT22

undiff.

HT22

diff.

0

10

20

30

40

50

60

70

80

90

100

s/

c

HT22 undiff.

HT22 diff.

Fig. 8. Time dynamics of the YO-PRO-1® (YP) uptake and its analysis. a - Time dynamics of the YP uptake of all four cell lines at 1.2 kV/cm, 8 × 100 μs, 1 Hz repetition frequency. One barmeans one standard deviation. Fluorescence f/arbitrary units is presented as a function of time t/min. b - The maximal value of YP fluorescence for all four cell lines at applied 8 × 100 μspulses of 1.2 kV/cm, 1 Hz repetition frequency. Value ofmaximal fluorescence f/arbitrary units is shown for each cell line. c - The resealing constantwhen 8×100 μs pulses at 1.2 kV/cmareapplied to all four cell lines. Value of time constant of resealing τ/s is presented for each cell line. Each error bar represents 95% confidence interval of thefittedoptimal value of the resealingconstant. The R-squared value was in all fits above 0.98.


10 ms. We modelled the strength-duration curve of excitable cells bythe Lapicque curve and the Hodgkin-Huxley model. We compared theexcitability results with permeabilization to 1 pulse of durations be-tween 150 ns and 100 ms, previously experimentally determined andmodelled using the Saulis model [67]. We also determined the perme-ability curve to 8 × 100 μs pulses, delivered at 1 Hz and different electricfield intensities. For the assessment of cell depolarization we used thePMPI dye. For the assessment of the plasma membrane perme-abilization we used the YO-PRO-1® (YP) dye. Thus, we assume all re-sults obtained with the dye PMPI to be depolarization due to voltage-gated channels opening or formation of the pores in the membrane.We assume all results obtained with the YP to be due to influx of YPthrough the voltage-gated channels or through permeable plasmamembrane. Ourmethodology does not allow us tomake distinction be-tween molecules entering through voltage-gated channels or throughpores/defects in the cell membrane.

The dye PMPI of the FLIPRR Membrane Potential Assay Kit is a valu-able tool for the measurement of membrane potential [78] and ionchannel pharmacology [79,80]. Although electrophysiology is consid-ered the gold standard for measurement of membrane potential, PMPIhas several advantages, namely the ease of use, the ability to monitorlong-term changes in multiple cells simultaneously. Furthermore,PMPI has been compared directly to electrophysiology data with agood agreement [68]. The quencher in the PMPI usually remains in theextracellular fluid due to its large size and continues to quench the fluo-rescence in the extracellular environment. However, when the plasmamembrane is permeabilized, the quencher could potentially enter thecell through pores formed in the membrane and decrease the fluores-cence in the cell. When we delivered pulses well above the depolariza-tion threshold, we observed a decrease in a fluorescent signal whichcould be indicative of cell electroporation. In the future, it should beestablished to what extent the PMPI dye could also serve as cell mem-brane permeabilization indicator.

The results of electrochemical depolarization show that with in-creasing K+ concentration, the fluorescence and thus the transmem-brane voltage are increasing which is in agreement with theory.However, it was unexpected that the highestfluorescencewas achievedwith CHO cells and notwith the differentiated HT22 cells. Excitable cellstypically have a higher density of voltage-gated channels, and thusmore ions should enter the cell when these channels are open. Resultsfor electrical depolarization to 1 ms and 0.6 kV/cm show similar trends– the highest change in fluorescence was measured with the CHO cells,and the response of the differentiated and non-differentiated HT22 cellswas similar (Fig. 5) which was surprising. CHO cells are non-excitablecells and would be expected to have a low background of voltage-

gated ion channels; however, some reports indicate that these cells ex-press voltage-gated Na+ channels [81] and a lack delayed rectifier K+

currents [82]. This would explain why CHO cells exhibited more signif-icant and prolonged depolarization in our experiments compared toother cells evaluated. Future investigations should examine the influ-ence of voltage-gated Na+ channel inhibitors on CHO depolarization.This finding highlights the importance of considering the endogenouscomplement of voltage-gated ion channels expressed in a given cellline It is also commensurate with our recent results using PMPI voltageimaging to demonstrate the numerous ion channels present and impli-cated in the depolarization of U-87MGcells by short electric pulses [83].Another difference between the CHO cells and other three cell lines istheir pattern of growth. CHO cells tend to form colonies and are inclose contact to one another which causes electric field shielding andlower induced transmembrane voltage [84]. Cells of U-87 MG andHT22 cell lines were grown to be more isolated (Fig. 1).

Our strength-duration curve was determined for pulse lengths be-tween 10 ns and 10 ms. As expected, the electric field needed to depo-larize cells decreased with increasing pulse duration [71,85,86]. In[86], the strength-duration curve was measured for frog muscles be-tween 1 ns and 100 ms. The authors determined that the thresholdsfor pulses of 100 μs or shorter followed a linear curve (in log-logscale) which confirmed that the signal was due to the opening of theionic channels and not due to electroporation. Our results had roughlythe same dependency but the linear curve followed a linear dynamicsfor pulses shorter than 1 μs. The differentiated HT22 cells are excitable[87] and because of similar shape of the strength-duration curve tothe curve by Rogers et al. [86] we can assume that for short pulses wealso measured predominantly opening of the voltage-gated channelsand notmembrane electroporation. The other three cell lines are not ex-citable, and the depolarization threshold only shows when the trans-membrane voltage was significantly increased above the restingvoltage. The reasons for the discrepancy between the data by Rogerset al. and our study could be that they used exponential pulses insteadof square pulses used in our study, and determined the threshold by ob-serving the twitching of the muscle while we used a fluorescent dye.

For the depolarization of the differentiated HT22 cells, higherelectric field intensity was needed at most pulse lengths. These re-sults are expected due to the more negative resting membrane volt-age of differentiated non-dividing cells [88] for which a higherchange in membrane potential is needed to reach the threshold ofdepolarization. However, at pulse lengths of 10 ns the differencein the critical membrane voltage between the differentiated HT22cells and non-excitable cells was up to 0.6 V (approximating cellsas prolate spheroids with R1 = 27 μm and R2 = 10 μm), which

109


cannot be explained solely by a lower resting transmembrane volt-age (around 50 mV difference between the excitable and non-excit-able cells). The endogenous voltage-gated channels present in theplasma membrane of the CHO cells could explain this observation.

First, we tried modelling the depolarization data with Lapicque'scurve which is similar to the Bleiss curve used in [86] but is valid forsquare pulses.We could not describe the shape of the strength-durationcurve well, and the optimized value of the chronaxie was higher thanpredicted from the value of the rheobase. Thus, we decided to fit theHodgkin-Huxley model, a more complex model, which takes into ac-count the dynamics of the voltage-gated channels. The Hodgkin-Huxleymodel in [71] couples the voltage on the anodal and cathodal pole of thecell via the equivalent circuit. We modelled the cell as an idealized pla-nar structure which does not accurately describe cell shape but it doesgive a general idea of the shape of strength-duration curve. Evenwhen calculating the strength-duration curve for a spherical cell, theshape of the strength-duration curve remained the same, it only slightlymoved along the y-axis (Fig. 7 in [71]).The shape of the strength-dura-tion curve is not linear. The Hodgkin-Huxley model qualitativelyfollowed the data well for pulses shorter than 10 μs, but for longerpulses, the predicted depolarization threshold decreased more thanthe experimentally determined one. The reasons for deviation at longpulses could be the following. First, we were also electroporating cellsenough for the dye to enter the cell but not the quencher. Second, oursystemwas not sufficiently sensitive to determine very low depolariza-tion thresholds. Third, for exact determination of the depolarizationthresholds or action potential (for the differentiatedHT22 cells) electro-physiological measurements are necessary. Fourth, also other channelscontributed to the dependency of depolarization observed in our exper-iments but were not included in the model.

In [52] the authors achieved excitation of a peripheral nerve with ananosecond pulse without electroporation. The presence of action po-tential without electroporation is in agreement with our modellingsince the thresholds for depolarizationwere lower than electroporationthresholds. Additionally, the nerves' refractory properties were notaffected.

We found the plateau of depolarization thresholds at 1 μs intriguingand decided to try the samemodel on bipolar pulses. The results of thismodel offered an interesting perspective on the potential mechanism oftheH-FIRE protocol [66]. In Fig. 6, we can see that around 1 μs there is anoverlap of the depolarization thresholds determined by the Hodgkin-Huxley model and of the Saulis permeabilization model. This overlapcould explain why with 1 μs bipolar pulses, electroporation was ob-servedwhile muscle contractionswere not [66]. In ourmodel, with cur-rently chosen parameters the threshold for electroporation was stillhigher than for an action potential, but the parameters of the Hodg-kin-Huxley model were not optimized to describe our data, and weused the same values as in [71]. Optimizing the values of the Hodgkin-Huxley model could bring curves closer together. The Hodgkin-Huxleymodel and permeabilization model also indicate that short monopolarpulses could be better at not exciting the tissues since the thresholdfor action potential was calculated to be higher than that of the bipolarpulses. A similar explanation why the H-FIRE pulses do not excite thecells was offered in [89] by numerical modelling of the response ofnerves to bipolar pulses. These authors showed that byusing short bipo-lar pulses, it was possible to ablate a tissue regionwithout triggering ac-tion potentials in the nearby nerve. The reason proposed being that thestimulation threshold rises faster than the irreversible electroporationthreshold. Further experiments are now needed to test these hypothe-ses, comparing the thresholds for action potentials and electroporationin excitable tissues with monopolar versus bipolar pulses in the 1 μsrange.

The repolarization time of all four cell lines was in the range of mi-nutes. The values for CHO, U-87 MG and non-differentiated HT22s arein agreement with the current knowledge existing as these cell linesdo not have IK voltage-gated potassium channels, responsible for fast

110

repolarization. The values of repolarization of differentiated HT22 cellsare much longer than traditionally observed during neural depolariza-tion,which is in the range ofmilliseconds. There are several possible ex-planations for this discrepancy. First, as the assessment method, weused PMPI dye entering the cell and then being pumped out. PMPI dyehas a time constant of several seconds [90] and can be understood as alow-pass filter. Second, our experiments were performed at room tem-peraturewhich slows down the speed of theNa/K pump. Third, it is pos-sible that delivery of a single pulse led to a burst of action potentials andthus a prolonged depolarization. Fourth, that cells were depolarized aswell as electroporated. Fifth, electroporation causes leakage of ATP[91] which is necessary for driving the pumps, and lack of ATP couldslow them down. Sixth, due to high induced transmembrane voltageion channels could be damaged [92,93].

The time required for reaching the peak fluorescence in depolariza-tion experiments coincidedwith the resealing time observed in the per-meabilization experiments. It is possible that during depolarization andpermeabilization experiments, PMPI and YP were entering through thevoltage-gated channels [94] as well as through pores formed in theplasma membrane. Even when using channel inhibitors, a total inhibi-tion of depolarization could not be achievedwhich indicates that duringdepolarization ions also enter through pores [83].

An interesting studywhere also a fluorescent dyewas used to assessdepolarization/action potential and plasma membrane electroporationof hippocampal neurons was recently performed by Pakhomov et al.[95]. They determined that the activation of voltage-gated sodiumchan-nels enhanced the depolarizing effect of electroporation. The authorsused the Fluo-Volt dye which enables imaging of fast changes in therange of ms. On the other hand, the PMPI dye enables imaging in therange of seconds to minutes, and it enables to capture slow, persistentchanges in the transmembrane potential in the non-excitable cells.

In the next part of our study, we exposed cells to 8 × 100 μs pulses,which are typically used in electrochemotherapy treatments. All fourcell lines reached the threshold of electroporation at approximatelythe same value - between 0.4 and 0.6 kV/cm. The permeabilizationcurve of all four lines could be described using a symmetric sigmoid. Al-though the differentiation causes a drop in the resting membrane po-tential [88], the lower, i.e. more negative resting membrane potentialdid not affect the threshold of electroporation as it was similar for excit-able and non-excitable cells. The permeabilization curves then followedsimilar dependency (Fig. 7b) and reached 50% of the maximal fluores-cence around 0.9 kV/cm.We can conclude that irrespective of the excit-ability, all four cell lines responded similarly to electroporation pulses.The results are in agreement with electroporation being a physicalmeans of disturbing plasma membrane in the lipid domain of themembrane. If the voltage-gated ion channels contributed to the YP up-take, it was much lower than the uptake through the permeabilizedmembrane.

The fluorescence reached was the highest for the U-87 MG cell lineand the lowest for the CHO cell line while the fluorescence of the differ-entiated and undifferentiated HT22 cells was between the levels of theCHO and U-87MG cells. Since the cells were grown attached in amono-layer, the lowest fluorescence of the CHO cell line can be explained bythe tendency of CHO cells to grow in colonies in proximity which de-creases the area of the plasma membrane available for dye uptake. An-other reason could be that the proximity of cells decreases the inducedtransmembrane potential due to shielding [76,84,96] although it is un-expected that the proximity did not affect the threshold of electropora-tion. YP starts to emit fluorescent light after binding to nucleic acids.Thus different concentration of intracellular nucleic acids could also af-fect the fluorescence.

From the time-lapse images of the YP uptake, the resealing speedcould be determined. The values of resealing constants were in thesame range as in [77] although the electroporation buffer we used inthis study was not tested previously. The cancerous U-87 MG cell lineresealed much faster than the other (normal) three cell lines which is

1511111

1511111

1511111

151111


in agreement with [97] where it was observed that cancerous cellsresealed 2–3 times faster than normal ones due to lower tension levelsin their lipid membranes.

6. Conclusions

In summary, the depolarization threshold was higher for the excit-able cells than for the non-excitable cells. The strength-duration curveof excitable cells was described with the Lapicque curve and the Hodg-kin-Huxley model. However, neither of the models described the be-haviour at all pulse durations. The Hodgkin-Huxley model gave insightinto the ability of the H-FIRE to permeabilize but not excite the tissue.All four cell lines responded similarly to pulses of standardelectrochemotherapy parameters. The shape of the permeability curvewas similar to curves already published in the literature [98]. Thus, elec-troporation is a feasible means of treating excitable and non-excitablecells with pulses of similar parameters. Furthermore, our results showthe potential of achieving permeabilization and minimizing or avoidingexcitation/pain sensation which needs to be explored in more detail. Infuture studies, it should be established, however, to what extent in vitrodepolarization and excitability correlate to the actual excitation andpain sensation in vivo.

Acknowledgement

Funding: This work was supported by the Slovenian ResearchAgency [research core funding No. P2-0249 and IP-0510, funding for Ju-nior Researchers to JDČ; the bilateral project Comparison of Electropo-ration Using Classical and Nanosecond Electric Pulses on Excitable andNon-excitable Cells and Tissues, BI-FR/16-17-PROTEUS], Region Limou-sin; CNRS; ANR Labex SigmaLim Excellence Chair and the Électricité deFrance Foundation ATPulseGliome project awarded to RPO. The re-search was conducted in the scope of the electroporation in Biologyand Medicine (EBAM) European Associated Laboratory (LEA).

The authors would like to thank D. Schubert (The Salk Institute, LaJolla, CA) for the kind gift of HT22 cells and X. Rovira (Institut deGenomique Fonctionnelle CNRS UMR5203, INSERM U1191, UniversityofMontpellier, France) for detailed cell culture protocols. JDČwould liketo thankD. Hodžić and L. Vukanović for their help in the cell culture lab-oratory and A. Maček-Lebar for help with the Hodgkin-Huxley model.

Conflict of interests

The authors have declared that no competing interests exist.

Authorship

JDČ: acquisition of the data, analysis and interpretation of the data,drafting the paper, final approval of the paper.

DM: conception and design of the study, analysis and interpretationof the data, drafting the paper, final approval of the paper.

MR: design of the ns-μs pulse generator, used in the study, final ap-proval of the paper.

PM: design of the ns-μs pulse generator, used in the study, final ap-proval of the paper.

SMB: acquisition of the data, analysis and interpretation of the data,final approval of the paper.

RB: acquisition of the data, analysis and interpretation of the data, fi-nal approval of the paper.

DAC: design of the ns pulse exposure setup, final approval of thepaper.

PL: analysis and interpretation of the data, design of the ns pulse ex-posure setup, final approval of the paper.

ROC: conception and design of the study, acquisition of the data,analysis and interpretation of the data, drafting the paper, final approvalof the paper.

Appendix A

Table A1Statistical parameters of the strength-duration curve statistical analysis by ANOVA for theCHO cell line (Table A1a), the U-87 MG cell line (Table A1b) the undifferentiated HT22(Table A1c) and the differentiated HT22 cell line (Table A1d). F(x,y) means the F-testvalue, where x, y are the degrees of freedom of the between-group comparison and thewithin-group comparison. Where Fw is indicated, the assumption of homogeneity of var-iance was not met; therefore, aWelch's adjustment for the ANOVAwas calculated and re-ported. The statistical significance provides the p-valueswhich are compared to our alphacriterion ofα=0.05. The effect size column provides an indication of the variability in re-sponse that can be attributed to the PEF. As an example,Ω2 = 0.47means that 47% of thechange in membrane potential can be attributed to the PEF.

Table A1a

CHO
F p
111

Ω2

0 ns
F(2, 17) = 9.00 p b 0.01 0.47 50 ns F(3, 15) = 4.56 p = 0.02 0.36 μs F(3, 16) = 91.52 p b 0.01 0.93 0 μs FW(3, 6.08) = 14.12 p b 0.01 0.67 00 μs F(4, 19) = 10.35 p b 0.01 0.61 ms FW(3, 5.55) = 53.76 p b 0.01 0.89 0 ms F(3, 13) = 10.80 p b 0.01 0.63
Table A1b

U-87 MG
F p Ω2
0 ns
F(4, 19) = 2.90 p b 0.01 0.80 50 ns F(3, 15) = 3.60 p = 0.04 0.29 μs F(3, 16) = 4.32 p = 0.02 0.33 0 μs FW(4, 9.27) = 43.54 p b 0.01 0.63 00 μs F(3, 21) = 47.64 p b 0.01 0.85 ms FW(4, 6.62) = 13.96 p b 0.01 0.71 0 ms F(3, 19) = 10.66 p b 0.01 0.56
Table A1c

Undifferentiated HT22
F p Ω2
0 ns
FW(3, 3.06) = 21.88 p = 0.01 0.81 50 ns F(2, 9) = 12.96 p b 0.01 0.67 μs F(3, 9) = 11.14 p b 0.01 0.70 0 μs F(4, 10) = 5.24 p = 0.02 0.53 00 μs F(2, 12) = 3.79 p = 0.05 0.27 ms F(3, 10) = 9.06 p b 0.01 0.63 0 ms F(4, 13) = 4.15 p = 0.02 0.41
Table A1d

Differentiated HT22
F p Ω2
0 ns
FW(3, 11.76) = 13.41 p b 0.01 0.60 50 ns F(2, 10) = 5.20 p = 0.03 0.39 μs F(2, 16) = 4.56 p = 0.03 0.29 0 μs F(6, 21) = 3.97 p b 0.01 0.39 00 μs F(4, 16) = 6.13 p b 0.01 0.49 ms F(3, 19) = 8.80 p b 0.01 0.50 0 ms F(4, 20) = 6.02 p b 0.01 0.45 1
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Paper 4 Title: Cell Electrosensitization Exists Only in Certain Electroporation Buffers

Authors: Janja Dermol, Olga N. Pakhomova, Andrei G. Pakhomov and Damijan Miklavčič

Publication: PLoS One, vol. 11, no. 7, p. e0159434, Jul. 2016.

DOI: 10.1371/journal.pone.0159434


Quartile: Q1 (Multidisciplinary Sciences)

Rank: 15/64 (Multidisciplinary Sciences)

115

RESEARCH ARTICLE

Cell Electrosensitization Exists Only in CertainElectroporation BuffersJanja Dermol1, Olga N. Pakhomova2, Andrei G. Pakhomov2, Damijan Miklavčič1*

1 Faculty of Electrical Engineering, University of Ljubljana, Ljubljana, Slovenia, 2 Frank Reidy ResearchCenter for Bioelectrics, Old Dominion University, Norfolk, Virginia, United States of America

*[email protected]

AbstractElectroporation-induced cell sensitization was described as the occurrence of a delayed

hypersensitivity to electric pulses caused by pretreating cells with electric pulses. It was

achieved by increasing the duration of the electroporation treatment at the same cumulative

energy input. It could be exploited in electroporation-based treatments such as electroche-

motherapy and tissue ablation with irreversible electroporation. The mechanisms responsi-

ble for cell sensitization, however, have not yet been identified. We investigated cell

sensitization dynamics in five different electroporation buffers. We split a pulse train into two

trains varying the delay between them and measured the propidium uptake by fluorescence

microscopy. By fitting the first-order model to the experimental results, we determined the

uptake due to each train (i.e. the first and the second) and the corresponding resealing con-

stant. Cell sensitization was observed in the growth medium but not in other tested buffers.

The effect of pulse repetition frequency, cell size change, cytoskeleton disruption and cal-

cium influx do not adequately explain cell sensitization. Based on our results, we can con-

clude that cell sensitization is a sum of several processes and is buffer dependent. Further

research is needed to determine its generality and to identify underlying mechanisms.

IntroductionElectroporation is a phenomenon resulting in a transient increase in membrane permeability,which occurs when short high voltage pulses are applied to cells and tissues [1,2]. If cells canrecover, we consider this reversible electroporation. If the damage is too extensive and they die,we consider this irreversible electroporation (IRE). Electroporation is used in medicine, e.g.electrochemotherapy (ECT) [3–6], non-thermal IRE as a method of tissue ablation [7–9], genetherapy [10,11], DNA vaccination [12,13] and transdermal drug delivery [14–16], as well as inbiotechnology [17], and food processing [18–20].

For tumor eradication, ECT, IRE, and gene therapy are successfully used. However, it wasshown that electroporation of tumors larger than 2 cm in diameter is not as successful as ofsmaller tumors [21]. When treating tumors with IRE, a high number of pulses is delivered,which can cause significant Joule heating and thermal damage and complicate the treatment[22,23]. Provided the effect of the electric pulses be enhanced, we can treat larger tumors with

PLOSONE | DOI:10.1371/journal.pone.0159434 July 25, 2016 1 / 19

a11111

OPEN ACCESS

Citation: Dermol J, Pakhomova ON, Pakhomov AG,Miklavčič D (2016) Cell Electrosensitization ExistsOnly in Certain Electroporation Buffers. PLoS ONE11(7): e0159434. doi:10.1371/journal.pone.0159434

Editor: Paul McNeil, Medical College of Georgia,UNITED STATES

Received: January 20, 2016

Accepted: July 1, 2016

Published: July 25, 2016

Copyright: © 2016 Dermol et al. This is an openaccess article distributed under the terms of theCreative Commons Attribution License, which permitsunrestricted use, distribution, and reproduction in anymedium, provided the original author and source arecredited.

Data Availability Statement: All relevant data arewithin the paper and its Supporting Information files.

Funding: This work was supported by the SlovenianResearch Agency (ARRS, www.arrs.gov.si) and byCOST TD1104 to JD, ref. no. STSM-TD1104-20915(www.electroporation.net). The research wasconducted in the scope of the electroporation inBiology and Medicine (EBAM) European AssociatedLaboratory (LEA). The funders had no role in studydesign, data collection and analysis, decision topublish, or preparation of the manuscript.

Competing Interests: The authors have declaredthat no competing interests exist.


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fewer pulses of lower voltage. Pulse effect amplification can be achieved using molecules thatenhance cell sensitivity to electric pulses, e.g. DMSO or surfactant C12E8 [8].

Lately, several reports on a so-called phenomenon of cell sensitization have been published[24–29]. By increasing the duration of an electroporation treatment (e.g. by decreasing thepulse repetition frequency or by splitting the delivered pulse train in more trains with delaysbetween them), a decrease in cell viability and a much higher uptake of molecules wereachieved. When applying conventional 100 μs pulses, 5 minute delay between the two trainswas suggested [25], but also shorter delays led to cell sensitization [27,30]. Cell sensitizationhas been observed as decreased membrane integrity, increased mass transport across the mem-brane, and decreased cell viability. Similarly as with square pulses, it has been shown thatexposing cells to AC electric fields increased their sensitivity to subsequent millisecond squarepulses [31].

Cell sensitization could be beneficially used in the electroporation-based treatments. It ispossible that it is already influencing the outcome of the IRE and the ECT. Namely, in the IRE,90 pulses synchronized with a heartbeat are delivered between each pair of electrodes [8]. Usu-ally, four electrodes are inserted, and IRE can last up to 9 minutes (four electrodes equals sixpairs, 6x90 pulses at around 1 Hz take 9 minutes). In ECT, eight pulses at 1 Hz or 5 kHz areapplied [32]. When using hexagonal electrodes, pulses are effectively delivered between 7 elec-trodes (12 pairs) [33]. Between each electrode pair, four pulses are delivered, and the procedureis repeated with four pulses of reverse polarity (twelve pairs, 8x12 pulses at around 1 Hz take1.5 minute). If we consider the switching time [34], both treatments already fall within the timerange for cell sensitization. The mechanisms of the delayed cell sensitization are not yetknown. The proposed mechanisms are: 1) calcium uptake [24,25], 2) ATP leakage [24,25], 3)reactive oxygen species formation [24,25], 4) cell swelling [24,25], 5) cytoskeleton disruption[28], 6) reduced pore edge line tension which lowers the electroporation threshold [26,27], 7)extended pore opening times [26,27], and 8) the decrease of high conductance membrane statewhich allows for the creation of additional defects [35].

We would like to emphasize the difference in the definition of the cell sensitization in thealready published studies and our paper. So far, cell sensitization has been defined as anincrease in total molecular uptake or decrease in cell survival after applying a split dose asopposed to a single dose. The contribution of separate pulse trains to the final uptake and sur-vival has not been investigated although, in our opinion, it is very important for the applicabil-ity of cell sensitization. When applying a single dose, we can reach saturation in mass transport[36,37]. Pulses at the end of the train contribute to the uptake less than pulses at the beginning[35,38]. With splitting the dose in half, we avoid saturation which is then mistakenly regardedas cell sensitization although cells respond to the first and the second pulse train in a similarway. In our experiments, we distinguished contributions of the first and the second pulse train.We considered the cells sensitized when the uptake due to the second pulse train was higherthan the uptake due to the first train, irrespective of the final fluorescence level.

In preliminary experiments, we tested cell sensitization in a standard low-conductivity elec-troporation buffer. Surprisingly, splitting the dose in half lowered the final fluorescence. Wewere intrigued and decided to test propidium uptake in the growth medium, where cell sensiti-zation has been previously observed [24,25,27]. There, splitting the dose in half increased thefinal fluorescence. We repeated experiments in three more electroporation buffers to investi-gate the effect of electrical conductivity, calcium influx, and sucrose concentration; splitting thedose in half again did not increase the final propidium uptake. Then, we analyzed the contribu-tions of the separate pulse trains to the final fluorescence. The response of cells to pulse split-ting in different electroporation buffers was complex and varied among others in the increasedor decreased sensitivity to the second pulse train, in the resealing speed, and in the efficiency of

Cell Electrosensitization in Different Electroporation Buffers

PLOS ONE | DOI:10.1371/journal.pone.0159434 July 25, 2016 2 / 19


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split versus single dose. Although cell sensitization has already been defined as a phenomenonand models have been constructed describing it [39], further research is needed to determineits generality and the involved mechanisms.

Materials and Methods

Cell preparationChinese hamster ovary cells (European Collection of Authenticated Cell Cultures ECACC,cells CHO-K1, cat. no. 85051005, obtained directly from the repository) were grown in 25 cm2

culture flasks (TPP, Switzerland) for 2–3 days in an incubator (Kambič, Slovenia) at 37°C andhumidified 5% CO2 in HAM-F12 growth medium (PAA, Austria). The growth medium wassupplemented with 10% fetal bovine serum (Sigma-Aldrich, Germany), L-glutamine (StemCell,Canada) and antibiotics penicillin/streptomycin (PAA, Austria) and gentamycin (Sigma-Aldrich, Germany). On the day of the experiments, the cell suspension was prepared. Cellswere detached by 2.5 ml of 10x trypsin-EDTA (PAA, Austria), diluted 1:9 in Hank’s basal saltsolution (StemCell, Canada). After 1.5 minute the trypsin was inactivated by 2.5 ml of thegrowth medium. Cells were transferred to 50 ml centrifuge tube and centrifuged 5 minutes at180g and 24°C. Then, cells were resuspended in the growth medium at 6x104 cells/ml. 500 μl ofcell suspension was added per well in 24-well plate. The plates were moved to the incubator for15–40 minutes until cells attached to the bottom of the well but maintained their sphericalshape [40].

Before the experiments, cells were washed with a fresh electroporation buffer and 500 μl ofthe new electroporation buffer was added to each well. For fluorescence measurements, theelectroporation buffer included 150 μM propidium iodide (Life Technologies, USA). The com-position of the tested electroporation buffers is given in Table 1. The electrical conductivity wasmeasured with conductometers MA 5959 (Metrel, Slovenia) or S230 SevenCompact (MettlerToledo, Switzerland) at 37°C and the osmolality by freezing point depression with Knauercryoscopic unit (model 7312400000, Knauer, Germany). MgCl2, NaCl, K2HPO4, CaCl2,HEPES, and sucrose were from Sigma-Aldrich, Germany, and KH2PO4 fromMerck, Germany

Electroporation and image acquisitionWe used Pt/Ir wire electrodes with 0.8 mm diameter and 4 mm inter-electrode distance posi-tioned at the bottom of the plate as shown in Fig 1. The plate was put under the microscopewith a heated stage (37°C). Square pulses (300V or 0.75 kV/cm, 100 μs, 10 Hz) were appliedusing the βtech electroporator (Electro cell B10, Betatech, France) and monitored with an oscil-loscope WaveSurfer 422, 200 MHz and a current probe AP015 (Teledyne LeCroy, Chestnut

Table 1. Composition of electroporation buffers.

Electroporationbuffer

Composition Electricalconductivity [mS/cm]

Osmolality [mOsm/kg]

The growth mediumHAM-F12

Inorganic salts, amino acids, vitamins and other components (PAA Austria,cat. no. E15-016), 10% fetal bovine serum, L-glutamine, and antibiotics

17.14 260–320 (based on theproducer’s data sheet)

The low-conductivitybuffer

10 mM KH2PO4/K2HPO4 in ratio 40.5:9.5, 1 mMMgCl2, 250 mM sucrose 1.78 292

The hyperosmoticbuffer

10 mM KH2PO4/K2HPO4, 1 mMMgCl2, 400 mM sucrose 1.76 475

The high-conductivitybuffer

10 mM KH2PO4/K2HPO4 in ratio 40.5:9.5, 1 mMMgCl2, 150 mM NaCl 19.12 300

The buffer with calcium 10 mM HEPES, 250 mM sucrose, 0.7 mMMgCl2, 0.3 mM CaCl2 0.38 281

doi:10.1371/journal.pone.0159434.t001




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Ridge, NY). In fluorescence measurements, we exposed cells to either half dose (four 100 μspulses), single dose (eight 100 μs pulses) or split dose (4+4, 100 μs pulses with 1, 2 or 3 minutedelay) (Fig 2). In the cell size measurements, we exposed cells only to the half dose (four 100 μspulses). 300 V was chosen based on the preliminary experiments, where most of the cells werepermeabilized, but the survival was not affected. 3 minutes was chosen as the delay that pro-duced cell sensitization in other studies [30] and where the measured final fluorescence valueof the split dose was higher (the growth medium) or lower (the low-conductivity buffer) thanof the single dose.

Cells were observed by the inverted microscope AxioVert 200 (Zeiss, Germany) under 20xobjective for fluorescence (535 nm excitation, 617 nm emission) and 40x for cell size measure-ments. Images were acquired using the VisiCam 1280 camera (Visitron, Germany) and theMetaMorph PC software (Molecular Devices, USA) in a time lapse: one image every 8 or 10 sfor 8 minutes (fluorescence measurements) or 2 s for 3 minutes (cell size measurements). Wechose a field of view in the middle between the electrodes where the electric field distributionwas approximately homogeneous, and electric field was calculated as a ratio of the applied volt-age to inter-electrode distance [41]. The numbers of the experiments and the analyzed cells foreach protocol are shown in Table 2. The exposure time in fluorescence measurements was dif-ferent across tested buffers, and the absolute values of fluorescence should not be directlycompared.

Images were analyzed using the ImageJ software (http://imagej.nih.gov/ij/). On fluorescentimages, each cell was manually outlined, and its average fluorescence intensity through thewhole stack was automatically measured. Fluorescence intensity before the pulse applicationwas subtracted to compensate for the background fluorescence. In preliminary experiments, itwas determined that in all tested buffers, there was no measurable spontaneous propidiumuptake on the same time scale without applying electric pulses. For cell size measurements, thethreshold was applied to the bright-field images, cells were automatically outlined, and their

Fig 1. Photo of the electrodes attached to the bottom of the 24-well plate. The plate was put under amicroscope. The images were acquired from between the electrodes where the electric field wasapproximately homogeneous and could be calculated as the ratio of the applied voltage and inter-electrodedistance.

doi:10.1371/journal.pone.0159434.g001




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http://imagej.nih.gov/ij/

area in each image was measured. The measured area was normalized to the size of the areabefore pulse application to determine a relative change in the cross-section. In each time step,the average value and the standard error were determined.

Fig 2. Scheme of the used protocols. In the half dose, four 100 μs pulses, and in the single dose, eight100 μs pulses were applied. In the split dose, two trains of four 100 μs pulses were applied and the delaybetween the end of the first and the beginning of the second train was 1, 2 or 3 min. In all experiments, 300 Vor 0.75 kV/cm were applied. The repetition frequency was 10 Hz.


Table 2. Number of experiments and analyzed cells in each experiment.

Type of experiment No. of experiments No. of analyzed cells

Fluorescence measurement The growth medium 4 pulses 3 19

8 pulses 5 34

4+4 pulses, 1 min delay 6 31



The low-conductivity buffer 4 pulses 5 34

8 pulses 6 45




The hyperosmotic buffer 4 pulses 4 21

8 pulses 3 20


The high-conductivity buffer 8 pulses 3 24


The buffer with calcium 8 pulses 3 21


Cell size measurement The growth medium 4 19

The low-conductivity buffer 7 25





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The model fittingThe quantitative results of the uptake were acquired by fitting a first-order model [42]:

f tð Þ ¼ C 1� exp � tt

� �� ð1Þ

to the experimental results using the Curve fitting toolbox and Matlab R2011b (Mathworks,USA). S signifies the plateau of the fluorescence; τ is the relaxation time of propidium uptakewhen the cells reach 63% of the final fluorescence. The time course of propidium uptakereflects the resealing process of the membrane and τ can also be understood as a resealing con-stant [42]. The derivation is in the S1 Appendix.

The model:

f tð Þ ¼ S1 1� exp � tt1

� �� þ S2 1� exp � t � tdelay

t2

� �� t > tdelay� �

þ kt ð2Þ

was fitted to the split dose results. Parameter tdelay is the inter-train delay, k is the slope of thelinear part of the propidium uptake. An example of the fitted curve and the parameters isshown in Fig 3. The first order kinetics model described the majority of our data sufficientlywell (R2>0.97), but it predicted that after the exponential closing of the pores the transport willstop, and the fluorescence will not increase. In our experimental results, the shape of the propi-dium uptake curves indicated that there was an additional process present that caused the lin-ear uptake. We modeled it with a linear uptake process (‘kt’) which we assumed present fromthe beginning of the treatment. Since we do not know what exactly caused the linear uptake, itseemed inappropriate to ascribe it only to one of the trains.

We fitted Eq 1 to the half and the single dose and Eq 2 to the split dose results. The optimalvalues of the parameters and the R2 (the coefficient of determination) value are presented inTable 3. In the split dose protocols, we normalized the S2 and τ2 values to the average S1 or τ1value in the same buffer (unless stated differently in Table 3). If S2/S1>1, cells were assumedsensitized, if S2/S1<1, they were assumed desensitized, and if S2/S1�1 the trains were assumedequally effective. Ratio τ2/τ1 gave information whether the resealing after the second train wasfaster (τ2/τ1<1) or slower (τ2/τ1>1) than after the first train or similar to the first train (τ2/τ1�1). The values of parameters S1 and τ1 should be the same for all first trains (half and splitdose) in each buffer. The difference in their values was due to the biological variability and the

Fig 3. An example of fitted first-order models to the results of propidium uptake in the growthmedium.Meaning of S1, S2, τ1 τ2, tdelay and the increase due to ‘kt’ are shown.





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Table 3. The optimal values of parameters and the R2 of fitted Eqs 1 and 2. A–The growth medium, B–the low-conductivity buffer, C–the hyperosmoticbuffer, D–the high-conductivity buffer, E–the buffer with calcium. Values in bold and the symbol next to them emphasize if the cells were desensitized (S2/average(S1)<1, symbol #), sensitized (S2/average(S1)>1, symbol ") or if the trains were equally effective (S2/average(S1)�1, symbol�). The numbersafter ± sign define 95% confidence interval. The graphical representation of S2/average(S1) and τ2/average(τ1), based on experimental uptake results in Fig 4,is in Fig 5.

A–The growth medium

Half dose Single dose 1 min delay 2 min delay 3 min delay

S1 215.5 ± 7.5 354.8 ± 9.9 179.6 ± 14.2 197.4 ± 17.2 204.2 ± 8.2

S2 / / 265.9± 13.9 226.9 ± 12.5 207.6 ± 8.8

τ1 65.91 ± 4.17 63.45± 3.33 44.02± 6.16 69.47± 10.07 46.13 ± 3.44

τ2 / / 48.37 ± 1.58 47.01± 3.25 37.76 ± 2.98

k 0.06664 ± 0.02 0.06825 ± 0.0267 0.1831 ± 0.0124 0.1234 ± 0.0339 0.1024 ± 0.0372

S2/average(S1) / / 1.34 ± 0.18 " 1.14 ± 0.15" 1.04 ± 0.13�τ2/average(τ1)

/ / 0.86 ± 0.20 0.83 ± 0.20 0.67 ± 0.16

R2 0.9954 0.9961 0.9998 0.9993 0.9991

B–The low-conductivity buffer

Half dose Single dose 1 min delay 2 min delay 3 min delay

S1 516.9 ± 3.9 965.7 ± 1.7 516.9 ± 3.9a 516.9 ± 3.9a 445.4 ± 19.4

S2 / / 413.1 ± 4.9 269 ± 12.4 229.2 ± 10.7

τ1 91.6 ± 2.66 82.78 ± 0.6 91.6 ± 2.66a 91.6 ± 2.66a 86.7 ± 4.97

τ2 / / 51.58 ± 3.15 65.11 ± 3.40 44.95 ± 3.65

k 0 0 0 0.0985 ± 0.0293 0.1512 ± 0.0556

S2/average(S1) / / 0.86 ± 0.04 # 0.56 ± 0.03# 0.48 ± 0.03#τ2/average(τ1) / / 0.58 ± 0.05 0.73 ± 0.06 0.50 ± 0.05

R2 0.9962 0.9997 0.9980 0.9994 0.9996

C–The hyperosmotic buffer

Half dose Single dose 3 min delay

S1 191 ± 2.4 387.7 ± 4.5 155.8 ± 27.3

S2 / / 86.76 ± 10.28

τ1 161.1 ± 5.3 117.8 ± 4.5 124.8 ± 21.2

τ2 / / 69.63 ± 12.46

k 0 0 0.01741 ± 0.0662

S2/average(S1) / / 0.50 ± 0.08 #τ2/average(τ1) / / 0.49 ± 0.10

R2 0.9977 0.9958 0.9992

D–The high-conductivity buffer

Single dose 3 min delay

S1 681.2 ± 9.1 369.8 ± 31.4

S2 / 379.8 ± 8.6

τ1 57.54 ± 1.54 122.2 ± 11.7

τ2 / 36.68 ± 1.34

k 0.4366 ± 0.0255 0.2719 ± 0.0623

S2/average(S1) / 1.03 ± 0.09�τ2/average(τ1) / 0.30 ± 0.10

R2 0.9992 0.9999

E–The buffer with calcium

Single doseb 3 min delay

S1 1129 ± 432 365.6 ± 22.4

S2 / 386.1 ± 22

τ1 144 ± 55.9 79.36 ± 11

(Continued)




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error of curve fitting. The error of curve fitting was more noticeable if the delay between twopulse trains was short and the first-order shape of the uptake curve was not very clear. There-fore, in order to decrease the error we presented results by normalizing the S2 and τ2 to theaverage values of S1 and τ1 for each buffer separately. In the cell (de)sensitization definition, ‘kt’was not included, since we assumed this additional process present from the beginning of themeasurement.

ResultsFirst, experiments were performed with a fixed inter-train delay (3 min) in five electroporationbuffers. Results are presented in Cell sensitization in different buffers section. Second, differentinter-train delays were tested in the growth medium and the low-conductivity buffer. Thesetwo buffers were chosen as a representative for the cell sensitization and cell desensitizationeffect. Results are presented in Cell sensitization with different delays section. The time dynam-ics of propidium uptake in the growth medium, the low-conductivity, the hyperosmotic, thehigh-conductivity buffer, and the buffer with calcium are shown in Fig 4A, 4B, 4C, 4D and 4E,respectively. The observations on higher or lower uptake as a response to the first or secondtrain are based on fitting Eqs 1 and 2 to the experimental results. One of the fitted curves isplotted in Fig 3 where the parameters S1, S2, tdelay, τ1, and τ2 are shown. In Table 3, the optimalvalues of parameters and the calculated ratios of S2/S1 and τ2/τ1 are presented. If S2/S1>1, cellswere assumed sensitized, if S2/S1<1, they were assumed desensitized, and if S2/S1�1 the trainswere assumed equally effective. The final level of fluorescence was irrelevant in our definitionof cell sensitization. In Fig 5A, there is the graphical representation of the ratios in the growthmedium, the low-conductivity, the hyperosmotic, the high-conductivity buffer, and the bufferwith calcium at fixed 3 minute inter-train delay. In Fig 5B, ratios for the growth medium, andin Fig 5C for the low-conductivity buffer at different delays are shown.

Cell sensitization in different buffersIn the growth medium, the total uptake after a split dose protocol (i.e. 4+4 pulses) was higherthan after a single dose protocol (i.e. eight pulses) at 1 minute delay (Fig 4A). The second traincontributed more to the total fluorescence (Fig 5A), i.e. cell sensitization was present. Thevalue of k was similar for the single and the half dose, but it was much higher for the split dose.There was a gradual uptake of propidium even 8 minutes after the pulses (k> 0).

In the low-conductivity buffer, the split dose protocol resulted in lower final fluorescencethan the single dose protocol (Fig 4B). In the split dose protocol, the uptake due to the secondtrain was lower than due to the first train (Fig 5A), i.e. cell sensitization was not present. On

Table 3. (Continued)

τ2 / 69.86 ± 5.54

k 0 0

S2/average(S1) / 1.06 ± 0.88�τ2/average(τ1) / 0.88 ± 0.14

R2 0.9705 0.9974

a The values for the S1 and τ1 were taken from the results of fitting Eq 2 to the results of the half dose (2nd column) where the plateau of fluorescence was

already reached, and the first order shape was clear.b Here, the first order model was not appropriate since the 95% confidence interval was very large. These parameters do not influence the results of the

analysis since the ratios were not calculated from the single dose parameters.





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the contrary, cell desensitization was present. The value of k was 0 for single, half dose and 1minute delay.

In the hyperosmotic buffer, the cells were exposed to half, single and split dose with 3 min-ute delay (Fig 4C). The split dose was less effective than the single dose. The uptake due to thesecond pulse train was lower than the uptake due to the first train (Fig 5C), i.e. cell desensitiza-tion was present. The parameter k was more than 0 only in the split dose protocol, and it was

Fig 4. Measured fluorescence due to the propidium uptake in the growthmedium and four differentelectroporation buffers. Propidium uptake was detected by fluorescence microscopy. On the y-axis, there are theaverage experimental values in arbitrary units ± standard error. On the x-axis, there is the time in seconds. (a), thegrowth medium; (b), the low-conductivity; (c) the hyperosmotic; (d), the high-conductivity buffer, (e) the buffer withcalcium. The exposure times were different among the electroporation buffers.





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10-times smaller than in the low-conductivity buffer (the same exposure time allows us tomake the comparison).

In the high-conductivity buffer, the cells were exposed to the single and split dose with 3minute delay (Fig 4D). The uptake due to the second train was similar to the uptake due to thefirst train (Fig 5C), i.e. cell sensitization was not present. The uptake due to the split dose was

Fig 5. Graphical representation of the ratio of parameters S2/S1 and τ2/τ1. A first order model (Eqs 1 and2) was fitted to the experimental results. S2 and τ2 were normalized to the average value of S1 or τ1,calculated from parameters of the half dose and the split doses with different delays. (a) The ratios at 3minute delay between the trains in all buffers. (b) The ratios in the growth medium at 1, 2 or 3 minute delay. (c)The ratios in the low-conductivity buffer at 1, 2 and 3 minute delay. The bars at 3 minute delay in (b) and (c)represent the same results as the growth medium and the low-conductivity buffer bars in (c) to enable easiercomparison of the ratios between the buffers. Error bars represent the 95% confidence interval.





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twice the uptake due to the half dose (determined by extrapolation). The k after the single dosewas higher than after the split dose.

In the buffer with added calcium, the final uptake due to the single dose was higher thandue to the split dose (Fig 4E). However, the uptake due to the second train was similar to theuptake due to the first train, i.e. cell sensitization was not present (Fig 5A). The value of k was 0for the split and single dose.

The values of τ after the second pulse train were lower than after the first pulse train in thegrowth medium and all electroporation buffers, i.e. the resealing after the second pulse trainwas faster than after the first train (Table 3). The values of τ after the first train were the lowestin the growth medium (44–69 s-1) and then they increased in the order: the buffer with addedcalcium (69–79 s-1), the low-conductivity (86–91 s-1), the high-conductivity (122 s-1), and thehyperosmotic buffer (124–161 s-1).

In the growth medium and the low-conductivity buffer, we also determined the cell sizechange after four 100 μs pulses (the half dose) (Fig 6). Cell size change was proposed as a possiblemechanism of cell sensitization [24]. In the growth medium, the cell cross-section did not changemuch. In the low-conductivity buffer, it decreased. In the low-conductivity buffer, the dynamicsof cell size change was different from the dynamics of the fluorescence change due to the secondtrain (Fig 4C). The uptake due to the second train was lower at 3 than at 2 minute delay while thecell cross-section did not change from the first minute until the end of the measurement.

Cell sensitization with different delaysIn the growth medium and the low-conductivity buffer, additional experiments with 1 and 2minutes delay between the trains were performed (Fig 4A and 4B). In the growth medium, thecell sensitization decreased with increasing inter-train delay (Fig 5B). At 3 minute delay, therewas no cell sensitization; the fluorescence was a sum of the separate contributions of the pulsetrains. The parameter k decreased with an increasing inter-train delay as well. In the low-con-ductivity buffer, we observed cell ‘desensitization’ (S2/S1<1) which increased with increasingdelay. The value of k was zero for single, half dose, and 1 minute delay.

Fig 6. Cell size change dynamics in the growthmedium and the low-conductivity buffer. Symbolsdenote the average ± standard error. The relative change was determined as a ratio of the cell cross-sectionafter and before the pulses. The upper magenta curve–the growth medium; the lower black curve–the low-conductivity buffer.





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DiscussionCell sensitization was not present in all tested electroporation buffers as determined by the pro-pidium uptake measurements. In the growth medium, the split dose was more efficient thanthe single dose at all delays, but the cells were sensitized by the first train only at 1 min. Inother buffers, the split dose was similarly (the high-conductivity buffer, and the buffer with cal-cium), or less effective (the low-conductivity, and the hyperosmotic buffer) than the singledose. In these buffers, thus cell sensitization was not present. After pulse application, differentprocesses are triggered in the cells or on their membranes. These processes apparently dependalso on the electroporation buffer composition and influence the response of the cells to theelectric pulses. The sum of these processes determines whether the cells will be sensitized,desensitized or the trains will be equally effective. In our study, if S2/S1>1, cells were assumedsensitized, if S2/S1<1, they were assumed desensitized, and if S2/S1�1 the trains were assumedequally effective. Cell sensitization, assessed by the uptake of different molecules or by survivalafter a single or a split dose protocol, was observed when applying pulses in the growthmedium [24,25,28,30], in electroporation buffers [24,25,43], in three-dimensional cell cultures[29], and in vivo [26,27]. Our results in the growth medium are in agreement with them, butthe results in the other buffers are not.

With our definition of cell sensitization (higher uptake due to the second than due to thefirst pulse train), the sensitizing effect was observed only at 1 minute delay in the growthmedium (Fig 5B). However, the final level of fluorescence in the growth medium was stillhigher when applying the split dose than when the single dose with all delays tested (1, 2 or 3min). In previously published studies, the higher final fluorescence of the split than of the singledose was already considered cell sensitization. There, cells were assumed sensitized if the splitdose caused higher uptake or lower survival than the single dose. With their definition, in ourexperiments in the growth medium, all delays would be considered as cell sensitizing althoughthe second pulse train did not cause higher uptake than the first pulse train, i.e. the cells werenot sensitized during the inter-train delay.

Pulse splitting in different buffersIn the growth medium, cell sensitization was present, and the split dose was more effective(Figs 4A and 5A) than the single dose. The reason could be higher membrane damage due tothe first pulse train than in other tested buffers which rendered cells more sensitive to the sub-sequent pulses. With increasing delay, the damage was repaired, and cell sensitization ceased.In other studies [24–27,30], the cell sensitization was present at longer delays. The possibleexplanations are: 1) They assessed cell sensitization as the efficiency of the split vs single dose.In our experiments, the final value of the fluorescence was higher when applying the split dose,although separate contributions to the final fluorescence of the two trains were equal. 2) Theydamaged the membrane more by applying more pulses of higher electric field. Since higherelectric fields cause higher cell sensitization [30], they could also cause longer cell sensitization.3) Our experiments were performed at 37°C where the membrane resealing is faster [44] whiletheir experiments were performed at room temperature.

In the low-conductivity buffer, there were protecting processes happening which take morethan 3 minutes to finish since the effect of ‘desensitization’ was still present with 3 minutesdelay. The time dynamics of propidium uptake was different than in the growth mediumwhich indicates that different mechanisms are involved. In the hyperosmotic buffer, the secondtrain of the split dose was even less effective than in the low-conductivity buffer. Since one ofthe differences between these buffers was sucrose concentration, we can assume that it has animportant role, as discussed in the following paragraphs. In the high-conductivity buffer, split




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and single dose caused similar final fluorescence (Fig 4G). Although sucrose inhibited cell sen-sitization, its absence did not cause it.

Time constant τ describes the time dynamics of the membrane resealing. Membrane reseal-ing is affected by several parameters. It was slower when the membrane was oxidized [45] andfaster at higher temperatures [44], lower sucrose concentration [46], and in the presence of lowCa2+ concentrations [42]. The cell membrane resealing depended on calcium, which had to bepresent in low concentrations when the cells were damaged either chemically, mechanically orelectrically [42,47–49]. The increased intracellular calcium concentration after electroporationserved as a stress indicator that triggered the membrane resealing process in the form of endo-[49], and exocytosis [47,48]. In the growth medium and the buffer with calcium, there werelow concentrations of Ca2+ present and τ was lower than in other buffers (faster resealing),probably due to Ca2+ facilitated membrane resealing. Even when calcium was not added, it wasstill present during electroporation. First, calcium is present in the intracellular storages [50].Second, even if it is not added to the buffer, there is some free calcium which comes from theglass or impurities in the chemicals, as measured in [45] for the low-conductivity buffer. Inter-estingly, in [47], the resealing after a second membrane disruption was much faster than afterthe first one. Authors suggested that initial wound was healed by exocytosis of the vesiclesfrom the endocytic compartment while the second wound resealed faster since Golgi apparatus(responsible for a formation of new vesicles) had already been activated by an increased Ca2+

influx. The role of Golgi apparatus is in agreement with our results since the resealing constantτ after the second pulse train was lower (faster resealing) than after the first one in all electropo-ration buffers (Table 3). Parameter τ decreased with increasing delay since there was moretime for the activation of the Golgi apparatus. The resealing speed after the half and the singledose was very similar in all tested buffers, contrary to what was observed by Rols and Teissié[51]. However, they applied more pulses (10 or 20 as opposed to 4 or 8 in our study) of higherintensity (1.2 kV/cm as opposed to 0.75 kV/cm in our study) which caused more pronouncedmembrane damage and difference in the resealing speed.

Parameter k describes the additional uptake process on a longer time scale. A higher valueof k indicates either higher final fluorescence or faster resealing since k ¼ S

t, as derived in the S1

Appendix. Thus, k depends on the final fluorescence and inversely depends of the resealingtime constant. In the growth medium, this additional process was always present. The value ofk decreased with an increasing delay, which suggests that either 1) the fluorescence due to theadditional process was decreasing or 2) its resealing speed was increasing. The decrease in thefinal fluorescence could be caused by the decreasing membrane permeabilization between thetrains. The second train caused less damage, and the additional uptake was lower than atshorter delays. In the low-conductivity buffer, the additional process of linear propidiumuptake was not always present. The trend of k was different than in the growth medium. Withincreasing delay, the final fluorescence of the additional process was increasing since slowerresealing is not likely (longer delay should decrease the resealing time). When the additionalprocess was not present, it finished too fast or was too small to be observed. With longer delays,the additional process was visible because it was not finished yet. Parameter k does not reflectthe binding delay of propidium after it enters the cells since it binds in microseconds [52].

In [43] pulses were split into two trains, and propidium or Yo-Pro-1 uptakes were measured.Interestingly, under the same pulse parameters, there was a large cell sensitization effect presentwith propidium, but much smaller with Yo-Pro-1 [53]. Apparently, different dyes give differentuptake results under the same conditions. Although pulses in the nanosecond range were applied,the observation is interesting also for the microsecond pulses since there are reports on cell sensi-tization with nanosecond [24] as well as with microsecond [25–27,30] pulses.




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Possible mechanisms of cell (de)sensitizationIncreasing sucrose concentration caused among others an increase in osmotic pressure whichdecreased the size or density of the permeabilizing structures in the membrane, slower mem-brane resealing [46] and stabilization of cell membrane [54]. Similarly, trehalose (a disaccha-ride similar to the sucrose) had a protective and stabilizing role [55,56]. Our results are inagreement with slower membrane resealing observation since τ increased with increasingsucrose concentration. Higher sucrose concentration also caused lower fluorescence due to thesecond pulse train—the cells were protected by a still unknown mechanism which is also inagreement with the protective role of the sucrose. When the sucrose was replaced by NaCl, thepulse trains were equally effective. Sucrose inhibits cell sensitization, but its absence does notfacilitate it.

The electrical conductivity of the growth medium and the high-conductivity buffer was inthe same range. In the high-conductivity buffer, there was no cell sensitization, while in thegrowth medium, cell sensitization was observed. Since a similar electrical conductivity inducesa different cell response, it is considered not to be responsible for cell sensitization.

In [35], authors ascribed cell sensitization to the higher efficiency of lower pulse repetitionfrequencies. Applying pulses on a permeabilized cell membrane is less effective because theexisting conducting structures prevent the establishment of an additional transmembranepotential, which they call ‘cell desensitization'. (The expression was used in a different sensethan in our paper.) Assuming that cell sensitization is a consequence of repetition frequencyeffect, one cannot explain such different behaviors in different buffers. Their explanation of cellsensitization effect is thus unlikely.

Extended pore opening times have also been proposed among possible mechanisms [26,27].The duration of all the protocols was the same; the membrane resealing was in the same timerange. Thus, the pore opening times were also of approximately the same duration. Theextended pore opening time does not explain cell sensitization.

Cell size change after electroporation occurs due to the colloid osmotic pressure [57,58].Electric pulses of different parameters cause different extent of cell permeability, and the cellmembrane becomes permeable to small molecules but not to large ones. The colloid osmoticpressure drives the influx of water and small solutes in the cell, and the cell swells [2]. Mole-cules, larger than the pores, balance colloid osmolality and stop cell swelling [58]. Increased/decreased cell size could cause higher/lower induced transmembrane voltage according to theSchwann equation. Cell size was measured via visible cross-section. In the growth medium, thecell size did not increase and thus cannot be the reason for cell sensitization. In the low-con-ductivity buffer, the cells shrunk. Cell desensitization could in part be explained by cell shrink-ing and lower induced transmembrane voltage. However, cell shrinking cannot be the onlyreason since the time dynamics of the uptake due to the second train and the cross-sectionchange were different.

Cell sensitization has been observed in several cell lines and in vivo. In our experiments, weused Chinese hamster ovary cells. Since in this cell line cell sensitization either was [25] or wasnot present, it does not depend only on the cell line. It is possible that even in the growthmedium cell sensitization would not be present in certain cell lines; however this remains to befurther investigated.

The cytoskeleton is damaged by electric pulses [59–61] and there were different observa-tions made (e.g. damaged cytoskeleton does not influence electroporation [62,63], it increasespermeabilization [28], it increases the survival [62], it changes the resealing dynamics [61]).Cytoskeleton disruption has also been connected to cell swelling [60]. In [28], the disruption ofthe cytoskeleton rendered cells more sensitive to electric pulses. However, the repair of the




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cytoskeleton is in the range of hours [59], but the cell sensitization dynamics in our experi-ments was in the range of minutes. The cytoskeleton disruption is not a likely explanation forcell sensitization.

The concentration of calcium we used in our buffer with calcium was similar as in theHAM-F12 growth medium. In the buffer with calcium, cells were more sensitive to the secondpulse train than in the low-conductivity buffer. However, this was due to the lack of sucrose inthe buffer with calcium. In the high-conductivity buffer where there is no calcium, the ratio S2/S1 was 1.03 and in the buffer with calcium, it was 1.06. In both buffers, the first and the secondpulse train were equally effective. It is possible that higher calcium concentrations would causecell sensitization but they could also lower cell viability [64]. Namely, electroporation with 1mM calcium already decreased survival of various cell lines to 20–40% [65].

In the literature, also other explanations for cell sensitization were suggested: ATP leakage,membrane oxidation, and reduced membrane line tension. The investigation of these factors isbeyond the scope of our current study. Our study opens many more questions which deservebeing investigated in future: the mechanism of sucrose’s protection of the membrane, the effectof different pulse parameters (duration and the number of pulses, electric field).

Modeling of cell electroporation is very useful since it allows us to describe cell responses,make predictions about effects of electric pulses on the cells, decrease the number of experi-ments needed, and help us to understand what is happening [37,39,66–68]. However, until alsobiological effects are included in the models, they will have a limited power and will not cor-rectly predict the outcome in certain cases.

Cell sensitization has also been observed in vivo by the delayed tumor growth, enhancementof tumor necrosis and perfusion defects [26,27]. The growth medium is an approximation ofthe in vivo extracellular fluid. The mechanism(s) for cell sensitization in vitro and in vivo couldbe similar. There are, however, many more factors in vivo–cell crowding, blood supply,immune system, electric field shielding.

ConclusionsCell sensitization seems to be a buffer dependent phenomenon. In the growth medium, stillunknown processes were triggered by the pulse application and rendered cells more susceptible toelectric pulses. In the low-conductivity and hyperosmotic buffer, a protective effect of the sucroseand cell shrinking caused cell ‘desensitization’. In the high-conductivity buffer, there was no cell‘desensitization’, probably due to the lack of sucrose. There was also no cell sensitization presentsince the cell ‘sensitizing’ processes from the growth medium were missing. The exact mechanismof cell sensitization has still to be determined; however we believe we shed additional light on theexisting hypotheses and discarded some of them. The effect of pulse repetition frequency [35], cellsize change, cytoskeleton disruption [24] and calcium influx [24] seem unlikely explanations.Based on our results, there are different sensitizing and desensitizing mechanisms present andcompeting at the same time, and the outcome depends on their overall contributions.

Supporting InformationS1 Appendix. Derivation of the first-order uptake equation.(DOCX)

AcknowledgmentsThe authors would like to thank prof. dr. Marija Bešter Rogač and Anton Kelbl from the Fac-ulty of Chemistry and Chemical Technology, the University of Ljubljana for their help with the




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http://www.plosone.org/article/fetchSingleRepresentation.action?uri=info:doi/10.1371/journal.pone.0159434.s001

osmolality measurements. JD would like to thank dr. Lea Rems for useful discussions onmolecular transport and acknowledge Lea Vukanović and Duša Hodžić for their help withexperimental work.

Author ContributionsConceived and designed the experiments: JD DMONP AGP. Performed the experiments: JD.Analyzed the data: JD DM. Wrote the paper: DM. Interpreted the results: JD DMONP AGP.

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S1 Appendix: Derivation of the first-order uptake equation

We can follow a similar derivation as (Shirakashi et al. 2002) and assume that after a pulse application cell

permeabilization decreases exponentially with time:

𝑃𝑃(𝑡𝑡) = 𝑃𝑃0 exp �−𝑡𝑡𝜏𝜏

�, (1)

where P(t) means the permeability at time t, P0 is the initial permeability at the end of the pulse and τ is the

resealing constant. Flux through the electroporated membrane can be written as in (Miklavčič and Towhidi

2010):

𝑗𝑗(𝑡𝑡) = 𝑃𝑃0 exp �−𝑡𝑡𝜏𝜏

� (𝑐𝑐𝑒𝑒 − 𝑐𝑐𝑖𝑖) ≈ 𝑃𝑃0 exp �−𝑡𝑡𝜏𝜏

� 𝑐𝑐𝑒𝑒 , (2)

where ci means the internal and ce the external concentration. We assume that all propidium in the cell binds

to nucleic acids immediately and that the intracellular propidium concentration is 0 mM. Namely, in

(Pucihar et al. 2008) it has been shown that the propidium signal starts to increase microseconds after the

pulse start. In our experiments, we measured the uptake in the range of seconds, and we could neglect the

time delay of propidium binding. Since in our experiments, we did not reach propidium saturation (the

fluorescence value of the treated cells was below the maximal measured fluorescence of electroporated

cells) we can assume that there was no unbound propidium in cells and thus the internal propidium

concentration was 0 mM. The number of molecules N that enter the cells can be obtained by integration of j

over time and permeabilized area ((Miklavčič and Towhidi 2010), Equation 9):

𝑁𝑁(𝑡𝑡) = 𝑃𝑃0𝜏𝜏𝑐𝑐𝑒𝑒𝐴𝐴𝑁𝑁𝐴𝐴(1− exp �−𝑡𝑡𝜏𝜏

�), (3)

where A is the permeabilized area and NA the Avogadro constant. By uniting τ, ce, P0, A and NA into a

constant C we obtain a first-order model:

𝑁𝑁(𝑡𝑡) = 𝐶𝐶 �1− exp �−𝑡𝑡𝜏𝜏

��. (4)

The fluorescence of propidium is linearly dependent on the number of the bound molecules below the

saturation level (Kennedy et al. 2008). The constant C does not have any meaning; it is only a multiplicative

factor and describes the plateau of the reached fluorescence. Thus, the fluorescence can be written as:

𝑓𝑓(𝑡𝑡) = 𝑆𝑆(1 − exp �− 𝑡𝑡𝜏𝜏 �), (5)

135


where f signifies the fluorescence in dependence on time and S is constant. The expression (5) can be

directly applied to our measurements.

The shape of the propidium uptake curves indicated that there is an additional process present that causes

the linear uptake seen after the first order process is finished. If we assume that this process is also first-

order and that its time constant is much larger than our observation time, we can use linear expansion and

include the first two terms:

exp(𝑡𝑡) = 1 +𝑡𝑡1!

+𝑡𝑡2

2!+ ⋯ = �

𝑡𝑡𝑛𝑛

𝑛𝑛!

∞

𝑛𝑛=0

. (6)

We obtain an approximation of Equation (5):

𝑓𝑓(𝑡𝑡) = 𝑆𝑆 �1− exp �−𝑡𝑡𝜏𝜏

�� ≅𝑆𝑆𝜏𝜏𝑡𝑡 = 𝑘𝑘𝑡𝑡,

(7)

where k equals the plateau of the process divided by the resealing constant.

1. Shirakashi R, Köstner CM, Müller KJ, Kürschner M, Zimmermann U, Sukhorukov VL. Intracellular Delivery of Trehaloseinto Mammalian Cells by Electropermeabilization. J Membr Biol. 2002;189: 45–54. doi:10.1007/s00232-002-1003-y

2. Miklavčič D, Towhidi L. Numerical study of the electroporation pulse shape effect on molecular uptake of biological cells.Radiol Oncol. 2010;44. doi:10.2478/v10019-010-0002-3

3. Pucihar G, Kotnik T, Miklavčič D, Teissié J. Kinetics of Transmembrane Transport of Small Molecules intoElectropermeabilized Cells. Biophys J. 2008;95: 2837–2848. doi:10.1529/biophysj.108.135541

4. Kennedy SM, Ji Z, Hedstrom JC, Booske JH, Hagness SC. Quantification of electroporative uptake kinetics and electricfield heterogeneity effects in cells. Biophys J. 2008;94: 5018–5027. doi:10.1529/biophysj.106.103218

136


Paper 5 Title: From cell to tissue properties – modelling skin electroporation with pore and local transport

region formation

Authors: Janja Dermol-Černe, Damijan Miklavčič

Publication: IEEE Transactions on Biomedical Engineering, vol. 65, no. 2, pp. 458-468, Feb. 2018.

DOI: 10.1109/TBME.2017.2773126


Quartile: Q1 (Biomedical Engineering)

Rank: 12/77 (Biomedical Engineering)

137

458 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 65, NO. 2, FEBRUARY 2018

From Cell to Tissue Properties—Modeling SkinElectroporation With Pore and Local Transport

Region FormationJanja Dermol-Cerne and Damijan Miklavcic

Abstract—Currentmodels of tissue electroporation eitherdescribe tissue with its bulk properties or include cell levelproperties, but model only a few cells of simple shapes inlow-volume fractions or are in two dimensions. We con-structed a three-dimensional model of realistically shapedcells in realistic volume fractions. By using a ‘unit cell’model, the equivalent dielectric properties of whole tis-sue could be calculated. We calculated the dielectric prop-erties of electroporated skin. We modeled electroporationof single cells by pore formation on keratinocytes and onthe papillary dermis which gave dielectric properties of theelectroporated epidermis and papillary dermis. During skinelectroporation, local transport regions are formed in thestratum corneum. We modeled local transport regions andincrease in their radii or density which affected the dielec-tric properties of the stratum corneum. The final model ofskin electroporation accurately describesmeasured electriccurrent and voltage drop on the skin during electroporationwith long low-voltage pulses. The model also accurately de-scribes voltage drop on the skin during electroporation withshort high-voltage pulses. However, our results indicate thatduring application of short high-voltage pulses additionalprocesses may occur which increase the electric current.Our model connects the processes occurring at the level ofcell membranes (pore formation), at the level of a skin layer(formation of local transport region in the stratum corneum)with the tissue (skin layers) and even level of organs (skin).Using a similar approach, electroporation of any tissue canbe modeled, if the morphology of the tissue is known.

Index Terms—Franz diffusion cell, local transport region,multiscale approach, numerical modeling, pore formation,skin electroporation.

I. INTRODUCTION

E LECTROPORATION is a physical way of disturbing cellmembrane with the application of short, high-voltage

pulses leading to increase in its permeability to differentmolecules (reversible electroporation) and to achieve cell death(IRE - irreversible electroporation) [1], [2]. Electroporation or

Manuscript received October 20, 2017; accepted November 8, 2017.Date of publication November 16, 2017; date of current version January18, 2018. This work was supported by the Slovenian Research Agency(research core funding nos. P2-0249 and IP-0510 and Junior Researcherfunding to JDC). (Corresponding author: Damijan Miklavcic.)

J. Dermol-Cerne and D. Miklavcic are with the Faculty of ElectricalEngineering, University of Ljubljana, Ljubljana SI-1000, Slovenia (e-mail:[email protected]).

Digital Object Identifier 10.1109/TBME.2017.2773126

pulsed electric field treatment is used in biotechnology [3], food-processing [4], [5] and medicine [6], e.g., gene electrotransfer[7], [8], DNA vaccination [9], [10], transdermal drug delivery[11], [12], IRE as an ablation technique [13], [14] and elec-trochemotherapy [15], [16].Transdermal drug delivery is easy to conduct, non-invasive,

quick and it avoids gastro-intestinal degradation. Skin is alsoa barrier not many drugs can penetrate, and various enhance-ment methods are used to improve drug delivery, electropo-ration being one of them [12]. However, skin electroporationis a complex phenomenon and still not well understood. It isbelieved that electric pulses cause the formation of regions ofan increased electrical conductivity through which the trans-port occurs – i.e., local transport regions (LTRs). The densityof the LTRs increases with increasing voltage while their sizeincreases with increasing pulse duration [17]–[22].Models of tissue electroporation enhance our understanding

as they offer a concise description using mathematical andphysical laws. The current tissue models are of two types – theymodel tissue as a bulk [23]–[26] while only few model tissues’microscopic structure [27]–[31]. The bulk tissue modelsdescribe the electric field distribution as a function of tissueproperties, applied voltage and the geometry of the electrodesbut assume that the tissue is homogeneous. For each tissue type,the critical electric field for electroporation or cell death must beexperimentally determined or described with statistical models[32]–[34]. Themodels with includedmicrostructure are in 2D or3D. The 2D models include sampling the tissue with Cartesiantransport lattices [30], [31], [35], describing it with the equiva-lent circuit [36], [37], or modeling it with the Voronoi network[29]. The 2D models disregard the geometry of the cells andcannot model different cell density which significantly altersthe equivalent dielectric properties, the induced transmembranevoltage and electroporation [38]. In 3D, tissuewasmodeled as aninfinite lattice [39], [40] or randomly distributed spherical cellsof different sizes [28]. The volume fraction maximally achiev-able by face-centered cube or hexagonal close-packed latticesis 77% which is lower than the volume fractions we encounterin most tissues (e.g., 83% in the epidermis, 91% in the stratumcorneum, 85% in the muscle tissue, 80% in the hypodermis[41]). Furthermore, cells were not of realistic shapes although itwas shown that cell shapes also affects the dielectric properties[42]. An improvement was a model of spinach leaf which also

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138

https://orcid.org/0000-0002-2555-9156

https://orcid.org/0000-0003-3506-9449

DERMOL-CERNE AND MIKLAVCIC: FROM CELL TO TISSUE PROPERTIES—MODELING SKIN ELECTROPORATION 459

included the size and shape of the cells; however, only a smallpart was modeled [27], and a generalization to the whole leafis questionable. The model by Huclova et al. [42] also modelssingle cells but offers a more realistic description since cells canbe of different shapes, sizes, and densities. Moreover, also tissueheterogeneity and anisotropy can be described which most ofthe current models cannot. The dielectric properties of cells aregeneralized to tissues as their equivalent dielectric properties.Numerical calculation is reduced to tissue level, andmuch largergeometry than a single cell can be modeled. The latter modeldescribes the behaviour of cells in the linear range but has a po-tential to include also nonlinear behaviour, i.e., electroporation.In the field of skin electroporation, only a few models exist

[20], [43]–[48] and they mostly belong to the group of bulktissue models and for the values of dielectric properties of sep-arate layers take rough approximations or even model severallayers with the same values. The values of the dielectric proper-ties of the electroporated layers are also rough approximations[44]. Some models include the formation of the local transportregions [43], [46], [48] but are focused only on the stratumcorneum and do not include pore formation in the membranesof the cells in lower layers. In the equivalent circuit of the skin[47], the information on the geometry of the cells is lost, andthe resistance of the elements is determined by experimentalmeasurements which vary between systems.Our aimwas thus to start from the microscopic scale to obtain

a bulk model of skin electroporation. We started with the modelby Huclova et al. [41] which has several advantages (possibil-ity to model real shapes and densities of cells, heterogeneityand anisotropy). We modeled the change in dielectric parame-ters of skin during electroporation and validated our model withcurrent-voltage measurements. We modeled pore formation incell membranes and local transport region formation in the stra-tum corneum. We modeled the skin’s dielectric properties aftertwo protocols which varied in voltage and pulse duration – longlow-voltage pulses (LV protocol) and short high-voltage pulses(HV protocol) [49].We achieved good agreement withmeasure-ments of LV protocol by including LTR formation in the stratumcorneum, change in electrical conductivity of the deeper skinlayers due to cell membrane electroporation and electrode po-larisation. During HV protocol the modeled voltage drop on theskin matched the measurements well; however, the current devi-ated for 50%. Also, the modeled density of the LTRs in the HVprotocol was much higher than the experimentally determinedvalues, which indicates that during the HV protocol additionalprocesses may occur which are not yet identified and were thusnot included in our model. The advantage of our approach isthat it is not limited only to skin electroporation, but electropo-ration of any tissue can be modeled provided the geometric anddielectric properties of cells constituting the tissue are known.

II. METHODS

A. Experimental Work

For fitting and validation of our model, we used experimentalresults from [49]. In short, 350 μm thick dermatomed porcineear skin was put between the donor and receiver chamber of the

Franz diffusion cell. Porcine ear skin is similar to the humanskin regarding layers’ thickness, number of hairs, their size andextension depth [50]. Two Pt electrodes were positioned 0.2 cmabove and 0.5 cm below the skin. Long low-voltage (3× 45Vpulses of 250 ms duration, 100 ms pause – LV protocol) orshort high-voltage pulses (3× 500V pulses of 500 μs dura-tion, 500 μs pause – HV protocol) were delivered to the skin.The delivered voltage, the voltage on the skin and current weremeasured.According to the model [41], the dermatomed skin consists

of four layers: the stratum corneum, the epidermis, the papillarydermis, and the upper vessel plexus.

B. Construction of the Model

The model of the skin before electroporation was constructedaccording to [41]. The modeling in our study aimed to add elec-troporation to the model by Huclova et al. The main steps inour ‘upgraded’ model are described by Fig. 1. We manuallychanged (i.e., optimized) the parameters in the local transportregion model and the electroporation in a unit cell model. Weobtained equivalent dielectric properties of separate layers andintroduced them into the Franz diffusion cell model. We appliedvoltage to the Franz diffusion cell model and calculated theelectric current and the voltage drop on the skin. We comparedthe simulated values with the measured ones and changed theparameters in the local transport region model and electropora-tion in a unit cell model until the measured and simulated valuesmatched. Values of dielectric and geometric parameters, used inthe final model, are referred to as ‘optimized values’ althoughthey could be only a local optimal values and not global.The numerical calculations were done in Comsol Multi-

physics (v5.3, Stockholm, Sweden) in 3D unless noted oth-erwise. All models were remeshed until there was a negligibledifference in the modeled voltage drop on the skin with furtherremeshing. In all models, electric currents physics was usedwhich calculates the electric potential in a subdomain by

−∇ (σ∇V) −∇jωε∇V = 0 (1)

where σ denotes the electric conductivity, ε the relative permit-tivity of the subdomain. jω in the frequency domain is equivalentto time differentiation in the time domain. Cell membrane wasmodeled by boundary condition distributed impedance, given by

n · J =1

dm(σm + jωε) (V − Vref ) (2)

where dm denotes the membrane thickness, ε its dielectricpermittivity, σm its electrical conductivity, Vref is the electricpotential on the exterior side of the boundary, n is the unit vectornormal to the boundary surface, J is the electric current density.

1) The Model of Dielectric Properties Without Electro-poration: We constructed the model as described in [41]. Formore detail on initial geometric and dielectric properties of thecells and settings of the numerical solver, we refer the reader to[41]. The authors modeled full thickness skin and also includedlayers which were not included in the dermatomed skin fromour experiments. Initially, we modeled all layers for validationof our model, but for modeling electroporation, only the stratum


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Fig. 1. Optimization scheme used in obtaining the parameters of the skin electroporation model. The parameters of the local transport regionmodel and the electroporation in a unit cell model were changed and introduced into the Franz diffusion cell model. In the Franz diffusion cell model,we simulated the current through the Franz diffusion cell and voltage drop on the skin. Results of the simulation were compared to the results of theexperiments. We varied the parameters of the models until the simulated and measured values matched.

corneum, epidermis, papillary dermis, and upper vessel plexuswere modeled. Briefly, the skin was divided into separate layers,based on theirmorphology. In each layer, a typical biological cell(i.e., corneocyte, keratinocyte,) or a typical structure (i.e., bloodvessels, collagen fibres) were identified. For the calculation ofequivalent dielectric properties, either analytical or numericalmethod was applied. When the particles were embedded in theextracellular fluid in a low volume fraction (<80%) the ana-lytical Hanai-Bruggeman equation was used in Matlab R2015b(Mathworks, USA) which was calculated much faster than thenumerical method. The papillary dermis was modeled as 40 μmspheres in 0.74 volume fraction and the upper vessel plexusas x-axis oriented cylinders with diameter 50 μm in volumefraction 0.3. Layers of higher volume fractions were modelednumerically. A unit cell was constructed for each cell layer. Abiological cell in a unit cell was modeled via the ‘superformula’,imported into the Comsol and scaled to achieve the correct sizesand volume fractions. These were (in x-, y- and z-direction withcorresponding volume fraction) for the corneocyte: (40 μm,40 μm, 0.8 μm, 0.85) and the keratinocyte: (5.97 μm, 5.97 μm,11.95 μm, 0.8). Several unit cells together could describe themorphology of the entire layer. Stratum corneum was modeledwith corneocyte, epidermis with keratinocyte and subcutaneoustissue with adipocyte. In Fig. 2, the corneocyte, the keratinocyteand the papillary dermis (used with the Krassowska asymptoticequation) are shown. The opposite boundaries of the unit cellwere exposed to a sinusoidal voltage, and via the admittance, thefrequency dependent dielectric properties were calculated (ε∗).Other outer boundaries were set to insulation. The process wasrepeated in all three axes to obtain the dielectric tensor (ε∗).From ε

∗, the electrical conductivity and relative permittivity

were calculated and inserted in the Franz diffusion cell modelas properties of separate layers. We used the frequency domainstudy, the direct PARDISO solver and the physics controlledmesh with finer or extra fine element size.

2) The Model of Dielectric Properties With Electropora-tion: We added skin electroporation to the model by Huclovaet al. [41]. We modeled pore formation on the level of singlecells as well as local transport region formation in stratum

Fig. 2. The geometry of the (a) corneocyte, (b) keratinocyte. Due to thesymmetry, only one-eighth was modeled. (c) Spheres in papillary dermiswere arranged in the face-centered cubic arrangement and could not bedescribed by including only one cell in the unit cell. Thus, for unit cell,we used a geometry which could periodically describe the geometry ofthe papillary dermis.

corneum. The pore formation was modeled on the keratinocytesand papillary dermis. The local transport region formation wasmodeled in the stratum corneum. The upper vessel plexus wasassumed not be electroporated.

a) The Model of Pore Formation: Pore formationwas included with Krassowska’s asymptotic equation [51], [52].We modeled pore formation on the keratinocytes [Fig. 2(b)] andspheres in the papillary dermis, [Fig. 2(c)]. The Krassowska’sasymptotic equation cannot be analytically modeled, and thuswe used the papillary dermis unit cell [Fig. 2(c)] and not theHanai-Bruggeman equation. Papillary dermis was modeled asspheres in the face-centered cubic arrangement in volume frac-tion 0.74. To include Krassowska’s asymptotic equation, weused the Weak Form Boundary Partial Differential Equationinterface with

dN

dt= αe

(I T VV e p

)2 (1 − N

N0e−q

(I T VV e p

)2 ), (3)


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TABLE IPARAMETERS USED IN UNIT CELL MODEL WITH KRASSOWSKA’S

ASYMPTOTIC EQUATION, THEIR MEANING, VALUE, AND REFERENCE

Symbol Meaning Value Reference

εe Dielectric permittivity of extracellular space(-)

80 [41]

σe Electrical conductivity of extracellular space(S/m)

0.53 [41]

εm Dielectric permittivity of cell membrane (-) 9.4 [41]σm 0 Initial electrical conductivity of cell

membrane (S/m)10−6 [41]

εi Dielectric permittivity of the intracellularspace (-)

50 [41]

σi Electrical conductivity of intracellular space(S/m)

0.12 [41]

dm Thickness of the cell membrane (nm) 7 [41]Vep Electroporation threshold (mV) 258 [52]α Electroporation parameter (m−2s−1) 109 [52]q Electroporation constant 2.46 [52]N0 Equilibrium pore density (m−2) 1.5× 109 [52]Rp Pore radius (nm) 0.75 [52]Vrest Resting membrane voltage (mV) −50 [52]σp Pore electrical conductivity (S/m) σ e −σ i

ln( σ eσ i

) [52]

σm Electrical conductivity of the electroporatedcell membrane (S/m)

σm 0 +σep

[52]

where N denotes the density of the pores in the membrane,and N0 in the unelectroporated membrane, ITV the inducedtransmembrane voltage, α, q, Vep describe the characteristics ofthe electroporation process [51], [52], and t is time. The poreformation caused an increase in electrical conductivity of thecell membrane:

σep = N2πr2

pσpdm

πrp + 2dm, (4)

where rp and σp were the radius and the internal electricalconductivity of a single pore, respectively, and dm is the cellmembrane thickness. The values of the parameters are in Table I.Increasing voltages were applied to the opposite boundaries ofthe unit cell in the z-direction. With increasing voltages, thedensity of the pores increased and consequently the inducedtransmembrane voltage. Dielectric tensor was calculated via theapplied voltage and the current flowing through the unit cell.In the experiments, the pulses were applied in the z-direction,and pore formation was therefore primarily in the z-direction.Electrical conductivity in x- and y-directions and the relativepermittivity in all three directions (x, y, z) were assumed un-changed and set to the unelectroporated values. We used thetime domain study, the direct PARDISO solver and the physicscontrolled mesh with finer or extra fine element size.

b) The Model of the Local Transport Region For-mation: The effect of electric pulses on the stratum corneumwas modeled via local transport region (LTR) formation. TheLTR was modeled as a cylinder in a unit cell (Fig. 3). We chosethe size of the LTR as stated it [49] and adapted the density bychanging the width and depth of the unit cell. In LV protocol,the LTR’s diameter was chosen according to [49] and in HVprotocol its density was adapted. Also here, we applied voltageon opposite boundaries in x- or z-direction, and via the electric

Fig. 3. Geometry of the local transport region (LTR) embedded in thestratum corneum (SC). The cylinder is the LTR. The outer region is theSC, unaffected by electric pulses. The image represents the size anddensity of the LTR at the end of low-voltage protocol at t = 1 s (aLV was443 μm and rLV was 90 μm).

TABLE IIGEOMETRIC AND DIELECTRIC PROPERTIES OF THE LOCAL TRANSPORTREGION AND THE UNELECTROPORATED STRATUM CORNEUM, THEIR

MEANING, VALUE AND REFERENCE

Symbol Meaning Value Reference/method

εLT R

(x, y, z)Dielectric permittivityof the LTR (-)

x and y:6.97× 104 ,z: 5.46× 102

Calculated with a unitcell withunelectroporatedcorneocyte

σLT R

(isotropic)Electrical conductivityof the LTR (S/m)

0.7 Optimized∗

εS C

(x, y, z)Dielectric permittivityof the SC (-)

x and y:6.97× 104 ,z: 5.46× 102


σS C

(x, y, z)Electrical conductivityof the SC (S/m)

x and y:1.38× 10−2 ,z: 2.29× 10−4


rLV Radius of the LTRduring LV protocol

Fig. 5(a) [49]

rH V Radius of the LTRduring HV protocol (μm)

10 [49]

aLV Length of the unit cellduring LV protocol (μm)

443 [49]

aH V Length of the unit cellduring LV protocol (μm)

177, 130,100, 80, 60,40, 30 (density inFig. 6(a))

Optimized∗

hLT R Height of the LTR (μm) 20 The same as the SCheight

hS C Height of the SC (μm) 20 [41], [49], [50]

SC denotes stratum corneum and LTR local transport region.∗Optimization was done by comparing measured and simulated current and voltage dropon the skin.

current calculated the dielectric tensor. Due to the symmetry,the properties in the x- and y-axis were the same. The geometricand dielectric properties of the stratum corneum and the LTRare in Table II. The dielectric properties of the electroporatedstratum corneumwere calculated under DC conditions since theapplied pulses in LV and HV protocol were relatively long andthe capacitive properties could be neglected. The size of the localtransport region was taken from [49] and slightly adapted as wetook into account that between pulses the LTRs could partiallyreseal which allowed a better description of our experimentaldata. We used frequency domain study, the direct PARDISOsolver and the physics-controlled mesh with extra fine elementsize.


141


Fig. 4. Geometry of the Franz diffusion cell with marked boundaryconditions and geometry. (a) the whole model, (b) zoomed to the areabetween the electrodes and (c) zoomed to the skin.

c) The Model of Franz Diffusion Cell: The Franzdiffusion cell model was built according to the measurementsand the data in [49]. InComsol, we used 2D axisymmetric geom-etry. Although the properties of the skin were anisotropic, therewas only 3% difference in the 2D and 3D calculation. However,the 3D calculation took approximately 6 hours and the 2D 1minute. The boundary conditions and the geometric parametersof the Franz diffusion cell model are marked in Fig. 4. Betweenthe electrodes, there was skin, immersed in phosphate buffer. Ontop of the electrodes, we applied the measured square voltagepulses. The other boundaries of the electrodes were assignedcontact impedance to model the impedance of the double layer,formed at the electrode-liquid interface:

n · J1 =1ρ

(V1 − V2) (5a)

n · J1 =1ρ

(V1 − V2) (5b)

where ρ denotes the surface resistance and V1 and V2 are thevoltages on each side of the boundary, n is the unit vector normalto the boundary surface, J is the electric current density. Prop-erties of the stratum corneum were obtained from the modelof the local transport region, properties of the epidermis fromthe unit cell of the keratinocyte and applied Krassowska’s equa-tion, properties of the papillary dermis from the unit cell ofthe spheres in the papillary dermis and applied Krassowska’sasymptotic equation. We used the physics-controlled mesh withnormal element size, time domain study and a direct MUMPSsolver.

III. RESULTS

After choosing the size [49] [LV protocol – Fig. 5(a)] andadapting the density [HV protocol – Fig. 6(a)] of the local trans-port regions (LTRs) we obtained the corresponding equivalentelectrical conductivity [Fig. 5(b) LV protocol and Fig. 6(b) HVprotocol] and relative permittivity [Fig. 5(c) LV protocol andFig. 6(c) HV protocol]. Fig. 5 thus shows results for the LV pro-tocol – in Fig. 5(a) we can see how the radius of the LTR changed

during the treatment, in Fig. 5(b) the change in the equivalentelectrical conductivity and Fig. 5(c) in equivalent relative per-mittivity in all three directions (x, y, and z). Fig. 6 shows resultsfor the HV protocol – on Fig. 6(a) we can see how the density ofLTRs changed during the treatment, in Fig. 6(b) the change inthe equivalent electrical conductivity and Fig. 6(c) in equivalentrelative permittivity in all three directions (x, y, and z).Fig. 7 shows the electrical conductivity of lower layers (papil-

lary dermis and epidermis) as calculated using the Krassowska’sasymptotic equation. The electrical conductivity varies with dif-ferent voltages on the boundary of the unit cell. Optimized val-ues of the electrical conductivity were inserted in the Franz,diffusion model. Figs. 8 and 9 show the final results of ourmodel – the simulated values of the voltage on the skin andthe current and are compared with the measured values. In theLV protocol (Fig. 8) the simulated and measured values of thevoltage drop on the skin [Fig. 8(a)] and the current [Fig. 8(b)]matched well while in the HV protocol [Fig. 9] the voltage dropon the skin corresponded well [Fig. 9(a)], but the current didnot [Fig. 9(b)]. The simulated current underestimated the actualmeasurement for approximately 50%.

IV. DISCUSSION

A. Experimental Considerations

In the stratum corneum (SC), the local transport regions(LTRs) were formed. In the epidermis (E) and papillary dermis(PD), cell electroporation occurred as SC resistance decreaseddue to LTR formation, and consequently, electric field increasedalso in the lower layers. Electric pulses increase the permeabilityof blood vessels and induce vascular lock [55], [56]. However,we assumed there was no electroporation in the upper vesselplexus (UVP) as it is a layer of blood vessels and even if weassumed electroporation to occur there is a lack of experimentaldata on its dielectric properties after electroporation.Change in dielectric properties of tissues can be used to de-

scribe and understand processes happening during electropo-ration. For assessing skin permeability, electric measurementswere used before [17], [57]–[60]. Dielectric properties can bemeasured by dielectric spectroscopy [61], current-voltage mea-surements during electroporation [62], electric impedance to-mography [63], and magnetic resonance electric impedance to-mography [64]. The efficiency of electroporation can be moni-tored by measurements of the change in electrical conductivity[25], [65], [66].Whenmeasuring dielectric properties of tissues,secondary effects can affect measurements – such as swellingof cells [67], forming of edema [65], loss of ions from the cells[62], vascular lock [55]. Thus, for the validation of our model,we used current-voltage measurements during pulse applicationwhen the secondary effects can still be largely neglected.Skin hydration and the electrical conductivity of the liquid in

which the skin is immersed was shown to have a large effect onskin impedance [68]. In our experiments, the dermatomed skinwas well hydrated as it was immersed in phosphate buffer. Wetook hydration into account by increasing the electrical conduc-tivity of LTRs to 0.7 S/m. This conductivity is in the same rangeas in [43] but is much higher than the electrical conductivity of


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Fig. 5. Dynamics of local transport region (LTR) during low-voltage (LV) protocol. (a) LTR size and the corresponding voltage on the skin, (b)electrical conductivity and (c) relative permittivity change. The obtained values were then taken as equivalent dielectric properties of the wholestratum corneum. The axes correspond to the coordinate system in Fig. 3.

Fig. 6. Dynamics of local transport region (LTR) during high-voltage (HV) protocol. (a) LTR density and the corresponding voltage on the skin,(b) electrical conductivity and (c) relative permittivity change. The obtained values were then taken as equivalent dielectric properties of the wholestratum corneum. Please, notice the time-scale which is only 300 μs although the whole treatment took 2.5 ms. We assumed that all changes inthe LTR density occur during the first pulse and afterwards there was no resealing or increasing of the LTR density. The axes correspond to thecoordinate system in Fig. 3.

Fig. 7. Electrical conductivity as a function of voltage, applied to a unitcell of a keratinocyte or papillary dermis. Even before pulse formation,the unit cell has a certain conductivity which then increases as a functionof applied voltage.

melted lipids (0.05 S/m [49]) which is relevant in the case ofdry skin.We assumed that the relative permittivity of the epidermis,

papillary dermis and upper vessel plexus do not change due toelectroporation since the applied pulseswere long and the capac-itive component of the layers at low frequencies is approximated

with an open circuit. The change in relative permittivity of thestratum corneum was based on the change in the geometry ofthe LTR.The buffer used in the donor compartment was phosphate

buffer, and in the receiver, it was phosphate buffered saline.They varied in the ionic composition and pH, but their electricalconductivity was the same (1 S/m). For our model, only thedielectric properties were important, and we modeled liquid inthe donor and receiver compartment as the same buffer.

B. Pore Formation in Keratinocytes and Papillary Dermis

The Krassowska’s asymptotic equation describes how thedensity of the pores changes as a function of applied voltage andtime. It is a simplification of a theory of pore formation as it as-sumes all pores in the cell membrane are of the same size whichis a justified approximation only for short nanosecond pulses.During longer pulses, the pores start to grow. For inclusion of anincrease in pore radius, creation of stable macropores should bemodeled [69], but the inclusion is computationally demanding.Therefore, we decided to calculate the change of electrical con-ductivity in the first few nanoseconds by Krassowska’s asymp-totic equation. As the sampling period of the current and voltage


143


Fig. 8. Measured and simulated (a) voltage on the skin and (b) currentthrough the system during LV protocol.

Fig. 9. Measured and simulated (a) voltage on the skin and (b) currentthrough the system during HV protocol.

measurements was much longer than nanoseconds, we assumedthat the electrical conductivity of the papillary dermis and theepidermis changed immediately when the pulse was applied.We calculated the dielectric properties at t = 0 s by the Kras-sowska’s asymptotic equation. To avoid modeling pore growth,we simplified our model and adapted the bulk electrical con-ductivity of separate layers at the end of pulse application. Thedielectric properties during pulse application were determinedby interpolation between t = 0 s and t = 1 s (LV protocol) ort = 2.5ms (HV protocol). The changes in electrical conduc-tivity of lower layers (papillary dermis and epidermis) had tobe included; otherwise we could not describe the experimentalmeasurements.The initial electrical conductivity of the epidermis and pap-

illary dermis was for the LV protocol approximated withKrassowska’s equation when 0.07 V or 0.08 V were applied tothe unit cell which corresponds to 69 mV or 71 mV of inducedtransmembrane voltage for corneocyte and papillary dermis, re-spectively. The initial electrical conductivity of the epidermisand papillary dermis was for the HV protocol approximatedwith Krassowska’s equation when 3 V were applied to the unitcell which corresponds to 1.14 V or 1.24 V of induced mem-brane voltage for corneocyte and papillary dermis, respectively.The initial theoretical voltage drop at t = 0 s on the epidermisand papillary dermis was determined by treating skin as a volt-age divider (right part of Fig. 10) on which there was a 30 V[LV protocol – initial peak of Fig. 8(a)] or 180 V [HV protocol– initial peak on Fig. 9(a)] voltage drop. Taking into account thegeometry (thickness of layers, area 0.785 cm2 [49], size of unitcells and corresponding number of unit cells in z-direction) and

Fig. 10. The equivalent circuit of the Franz diffusion cell. The electrodepolarisation is represented only by resistance. Since we are deliveringrelatively long pulses, the capacitance due to a double layer which is inparallel with Relectrodes , could be approximated as an open circuit andthus neglected. Rsk in is represented by separate resistances of eachlayer. The voltage drop on the skin is thus distributed among RSC , RE ,RPD and RUVP , according to the thickness and electric conductivity ofeach layer.

initial electrical conductivity of each layer before electropora-tion in z-direction (SC: 2.29× 10−4 S/m, E: 6.36× 10−2 S/m,PD: 7.19× 10−2 S/m, UVP: 3.85× 10−1 S/m) we calculatedthe resistance of each layer and a corresponding voltage dropon each layer. This calculated voltage drop was then used in theKrassowska’s asymptotic model.With HV protocol, the initial theoretical voltage drop on the

epidermis and papillary dermis was 0.4 V, but a good agreementbetween measurements and model was achieved with 3 V. Theinitial electrical conductivity was thus higher than theoreticallypredicted. Higher initial electrical conductivity indicates thatsome defects were formed in the stratum corneum quickly andlower layers were exposed to a higher voltage than predicted.In HV pulses, the change in electrical conductivity was the

largest during the first pulse, and it changed only slightly dur-ing the following two pulses. Between pulses, there was somedecrease in electrical conductivity present which indicates re-sealing of the cell membranes as defects in the stratum corneumwere reported to reseal on a timescale of ms - hours [22].

C. Local Transport Region Formation in the StratumCorneum

It was shown that short high-voltage pulses increase the den-sity of the local transport regions (LTRs) while low-voltagelong pulses increase the size of LTRs by Joule heating and lipidmelting around the pre-existing defects [17]–[22]. The radiusof the LTR during LV protocol increased from 10 μm to 90 μmand was taken from [49] and the values are within the rangesreported by [21], [22]. We assumed that LTRs reseal to someextent between pulses [Fig. 6(a)] although as with the HV pro-tocol, it is possible that cells in the lower layers resealed as well.Since the pause during LV protocol was longer than during HVprotocol (100 ms vs 500 μs), we modeled LTR resealing insteadof corneocyte or papillary dermis resealing. The time courseof recovery in skin electroporation ranges from a few ms up to


144


several hours [22]. Thus, we assumed that it is possible that LTRradii slightly decrease during the pause between pulses. Still, thisis only an assumption – the increase in the resistance of the skincould as well be a consequence of resealing of the pores in themembranes of the cells in the lower layers. We have added theresealing in ourmodel, and by doing so, wewere able to describethe measurements of voltage and current more accurately.The increase in electrical conductivity of SC during HV pro-

tocol was modeled by an increase in density of the LTRs whiletheir radii stayed the same (10 μm). The modeled density of theLTR-s was high (up to 1200 LTRs/mm2) in comparison to previ-ously published studies ([43] and the references therein) (up to9 LTR/mm2). We tried decreasing the density of the LTRs downto the reported values, but the modeled voltage on the skin was2-times higher than measured which indicates that with lowerLTR density the predicted electrical conductivity of the skin istoo low. Even with increasing the electrical conductivity of epi-dermis and papillary dermis to 1 S/m (electrical conductivity ofthe phosphate buffer), the voltage on the skin did not decreaseenough to obtain voltage on the skin that would correspond tothe measured values. This indicates that LTR density is mostprobably higher than expected.

D. Measurements and Simulation in the Franz DiffusionCell

An equivalent circuit can represent the Franz diffusion cell(Fig. 10), where Relectrodes , RP BS and Rskin denote the resis-tance of the double layer on the electrodes, phosphate bufferand skin, respectively. Udel is the voltage, delivered to the elec-trodes, and Uskin the voltage, measured on the skin. The doublelayer acts as an additional resistance in parallel to capacitance[53]. Since our pulses were relatively long, we assumed the ca-pacitance as an open circuit, and only the resistance Relectrodeswas included in the equivalent circuit (Fig. 10). This additionalresistance caused a drop of voltage on the electrodes, and thesample received less voltage than what was delivered to elec-trodes. In the case of LV protocol, the voltage drop on electrodeswas significant in comparison to the applied voltage and voltagedrop on the skin (around 1 V on the electrodes with delivered45 V and 12–30 V on the skin). In the case of HV protocol,the voltage drop on the electrodes did not influence the resultsbut was included for consistency. Relectrodes , RP BS , Udel , andUskin were known, which means that according to the voltagedivider, Rskin is already determined:

Rskin =Uskin (Relectrodes + RP BS )

Udel − Uskin(6)

The simulated voltage drop on the skin and current throughthe system after the LV protocol corresponded well with themeasurements (Fig. 8). The fitting was obtained by adaptingthe electrical conductivities of separate layers until a goodagreement was obtained between simulated and measuredvalues. Taking into account that in LV protocol, the simulatedand measured current match [Fig. 8(b)], our model was thusadditionally validated.

Fig. 11. Position of the measuring Cu electrodes. When in parallel, theelectrodes were not directly one on top of another as they would squeezethe skin and change the thickness of the skin. When the electrodes werecrossed, there was a preferential path for electric current.

In HV protocol, however, when we achieved a good agree-ment between simulated and measured voltage drop on the skin,the simulated and measured current did not match [Fig. 9(b)].One possibility is additional processes, e.g., introduced by mea-suring Cu electrodes. Namely, when measuring the voltage dropon the skin, theCu electrodeswere in contactwith skin. The elec-trodes were put on the skin between the donor and receiver partof the Franz diffusion chamber in two possible configurations:parallel and crossed (Fig. 11) (M. Rebersek, personal commu-nication). Unfortunately, the exact position of the measuringelectrodes was not noted during experiments and is thus notknown. Crossed configuration introduced a preferential path forelectric current since Cu is much more conductive than PBS orskin. It possible that there was localised heating and dischargespresent which caused the current to increase significantly, lo-cally thus contributing to the overall current measured, but notcausing a detectable voltage drop.The shape of the voltage and current measurements reflects

the different phenomena occurring during electroporation. Theinitial voltage drop and increase in current in the first few mil-liseconds is due to LTR formation in the stratum corneum. Theemergence of LTRs increases the electric conductivity, but withincreasing the radius of the LTRs, the effect slowly reaches aplateau. With decreasing resistance of the SC, the electric fieldincreases in the lower layers of the skin. The prolonged decreasein voltage and increase in current is due to a slow but constantincrease in the electric conductivity of lower layers due to poreformation in the cell membranes of the cells in the lower layers,i.e., electroporation.From Table III we can see that HV protocol increased the

electrical conductivity of all layers more than the LV protocol.In LV protocol, the delivered voltage was 45 V, and on the skin,it was 12 V–30 V (26%–66% of the delivered voltage). In HVprotocol, the delivered voltage was 500 V, and on the skin, itwas 100 V–180 V (20%–36% of the delivered voltage), whichis much less than what was delivered to the skin in the LVprotocol.When electrodes are immersed in liquid, a double layer is

formed at the electrode-liquid interface. The surface resistancedue to the double layer changes with current density, electrodematerial, ions in the liquid. Interestingly, although it can sig-nificantly change the voltage the cells and tissues are exposedto, it is included only in a few models [70]. We used the values


145


TABLE IIIGEOMETRIC AND DIELECTRIC PARAMETERS OF THE FRANZ DIFFUSION CELL

MODEL, THEIR MEANING, VALUE AND REFERENCE

Parameter Meaning Value Reference/Method

w Radius of the skin sample (cm) 0.5 [49]hsc Height of SC (μm) 20 [41], [49], [50]he Height of E (μm) 100 [41]hpd Height of PD (μm) 140 [41]huv p Height of UVP (μm) 80 [41]dd Distance between electrode in the

donor solution and skin (cm)0.2 [49]

dr Distance between electrode in thereceiver solution and skin (cm)

0.5 [49]

T Temperature of the FDC (°C) 37 [49]wpla te Radius of the plate on the electrode

(mm)3.5 [49]

wel Radius of the electrode (mm) 0.5 [49]h Height of the FDC (cm) 5.255 Measuredρs Surface resistance (Ωcm2) 3.03 [53]εP B S Dielectric permittivity of the

PBS(-)80 [52]

σP B S Electric conductivity of the PBS(S/m)

1 [49]

σel Electric conductivity of the Ptelectrodes (S/m)

9.43× 106 [54]

εS C , LV

(x, y, z)Dielectric permittivity of the SCafter LV protocol (-)

Fig. 5(c) Optimized∗

σS C , LV

(x, y, z)Electric conductivity of the SCafter LV protocol (S/m)

Fig. 5(b) Optimized∗

εS C , H V

(x, y, z)Dielectric permittivity of the SCafter HV protocol (-)

Fig. 6(c) Optimized∗

σS C H V

(x, y, z)Electrical conductivity of the SCafter HV protocol (S/m)

Fig. 6(b) Optimized∗

εE

(x, y, z)Dielectric permittivity of the E (-) x and y:

4.83× 103 , z:1.52× 104

The unit-cellmodel withkeratinocyte

σE , LV

(x, y, z)Electrical conductivity of the Eafter LV protocol (S/m)

x and y:5.82× 10−2 ,z: 6.36× 10−2

at t = 0 and7.69× 10−2

at t = 1 s

Optimized∗

σE , H V

(x, y, z)Electrical conductivity of the Eafter HV protocol (S/m)

x and y:5.82× 10−2 ,z: 0.12 att = 0 and0.17 att = 2.5ms

Optimized∗

εP D

(isotropic)Dielectric permittivity of the PD (-) 1.62× 104 The Hanai-

Bruggemanequation

σP D , LV

(isotropic)Electrical conductivity of the PDafter LV protocol (S/m)

0.086 att = 0, 0.096at t = 1 s

Optimized∗

σP D , H V

(isotropic)Electrical conductivity of the PDafter HV protocol (S/m)

0.13 at t = 0,0.18 att = 2.5ms

Optimized∗

εU V P

(x, y, z)Dielectric permittivity of theUVP (-)

x: 4.92× 104 ,y and z:6.40× 104 ,

The Hanai-Bruggemanequation

σU V P

(x, y, z)Electrical conductivity of the UVP(S/m)

0.42, 0.39,0.39

The Hanai-Bruggemanequation

(x, y and z) denote the anisotropic properties in x-, y- and z-direction. SC denotes stratumcorneum, E epidermis, PD papillary dermis, UVP upper vessel plexus, PBS phosphatebuffer, and FDC the Franz diffusion cell. Dielectric permittivity of material was not relevantas it did not influence the results since the pulses were relatively long.∗Optimization was done by comparing measured and simulated current and voltage dropon the skin.

from a study [53] where they used Pt electrodes. However, theymeasured resistance in saline instead of phosphate buffer, andtheir electrodes were smaller, which could affect the results.More concerning is that they delivered much lower voltage thanthe voltage delivered in our experiments. The chemistry of highvoltage pulses delivered to an electrolyte is still not well under-stood [71], with few studies focusing on the release of metalfrom electrodes [72] and possible reactions happening on ornear the electrodes [73], [74].The drawback of our model is that there are several com-

binations of parameters which can offer good agreement withthe current and voltage measurements – e.g., we could changethe dynamics of local transport region formation and describethe exponential decrease of voltage drop on the skin with dif-ferent dynamics of change of electrical conductivity of lowerlayers. We could compensate for the different geometry of theLTRs with a different electric conductivity of the LTR. Themodel, obtained in this study, is not trying to replicate the ex-perimental results precisely but serves as a proof of principlehow cell level electroporation can be expanded to tissue levelelectroporation. For a more precise determination of parame-ters, more experiments and electrical measurements are neededas the values of several parameters are currently only estimated(for example, dielectric properties of separate layers duringelectroporation).The advantage of using our model is that we can use it to

describe any tissue with any resolution we want, we can includeblood vessels, lymphatic system, different types of cells in a tis-sue etc. The only requirement is that the geometric and dielectricproperties of the tissues’ microstructure are known. When gen-eralizing the properties to a tissue level, the computations arequick and enable inclusion of other physical phenomena – forexample, heating or chemical reactions.

V. CONCLUSION

Our model connects the processes occurring at the level ofcell membranes (pore formation), and at the level of a skin layer(formation of local transport region in the stratum corneum),with the level of tissue (skin layers) and even level of organs(skin). It enables description and prediction of changes in dielec-tric properties of tissue while taking into account microstruc-ture and processes on a much smaller scale and simultaneouslykeeping the computational times reasonable. Not just skin butany other tissue could be described similarly, as long as its mi-crostructure is known. Our model thus offers a step forward inmodeling of electroporation at different spatial scales.

ACKNOWLEDGMENT

J. Dermol-Cerne would like to thank Dr. Huclova and Dr.Frohlich for their help with reconstructing the original model,and Dr. Kos and Dr. Rems for their help with Comsol modeling.The research was conducted in the scope of the electropora-tion in Biology and Medicine (EBAM) European AssociatedLaboratory (LEA).


146


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148


Paper 6 Title: Connecting the in vitro and in vivo experiments in electrochemotherapy: Modelling cisplatin

transport in mouse melanoma using the dual-porosity

Authors: Janja Dermol-Černe, Janja Vidmar, Janez Ščančar, Katja Uršič, Gregor Serša, Damijan Miklavčič

Publication: Journal of Controlled Release, submitted.


149


Connecting the in vitro and in vivo experiments in

electrochemotherapy: Modeling cisplatin transport in mouse

melanoma using the dual-porosity model

Janja Dermol-Černe1, Janja Vidmar2, Janez Ščančar2, Katja Uršič3, Gregor Serša3,4, Damijan Miklavčič1

1University of Ljubljana, Faculty of Electrical Engineering, Tržaška cesta 25, 1000 Ljubljana, Slovenia

2Jozef Stefan Institute, Department of Environmental Sciences, Jamova cesta 39, 1000 Ljubljana, Slovenia

3Institute of Oncology Ljubljana, Department of Experimental Oncology, Zaloška cesta 2, 1000 Ljubljana,

Slovenia

4University of Ljubljana, Faculty of Health Sciences, Zdravstvena pot 5, 1000 Ljubljana

Corresponding author: Damijan Miklavčič ([email protected])

Declarations of interest: none

150


Abstract In electrochemotherapy two conditions have to be met to be successful – the electric field of sufficient

amplitude and sufficient uptake of chemotherapeutics in the tumor. Current treatment plans only take into

account critical electric field to achieve cell membrane permeabilization. However, permeabilization alone

does not guarantee uptake of chemotherapeutics and consequently successful treatment. In our study, we

described the transport of cisplatin in vivo based on experiments performed in vitro. In vitro, a spectrum of

parameters can be explored without ethical issues. In the experimental part of our study, we performed in

vitro and in vivo experiments. Mouse melanoma B16-F1 cell suspension and inoculated B16-F10 tumors

were exposed to electric pulses in the presence of chemotherapeutic cisplatin. The uptake of cisplatin was

measured by inductively coupled plasma mass spectrometry. In the modeling part of our study, we modeled

the transport with the dual-porosity model which is based on the diffusion equation, connects pore

formation with membrane permeability, and includes transport between several compartments. In our case,

there were three compartments - tumor cells, interstitial fraction and peritumoral region. Our hypothesis

was that in vitro permeability coefficient can be introduced in vivo, as long as tumor physiology is taken

into account. A transformation from in vitro to in vivo was possible by introducing a transformation

coefficient which takes into account in vivo characteristics, i.e., smaller available area of the plasma

membrane for transport due to cell density, and presence of cell-matrix in vivo, reducing drug mobility. Our

model offers a step forward to connecting transport models at the cell level to the tissue level.

Keywords: electrochemotherapy, drug delivery, transport, modelling, dual-porosity model

1. Introduction When biological cells are exposed to short high-voltage pulses, the permeability of the cell membrane

increases, presumably due to pore formation following electroporation. As a result, membrane-impermeant

molecules can pass the membrane [1–4], presumably through newly formed pores. If cells recover, it is

called reversible electroporation. If the damage is too severe and cells die, it is called irreversible

electroporation. Electroporation is used in biotechnology [5], food processing [6,7] and medicine [8,9], e.g.

electrochemotherapy [10], gene electrotransfer [11,12], irreversible electroporation as an ablation technique

[13,14] and transdermal drug delivery [15,16].

In electrochemotherapy transport of chemotherapeutics is critical for a successful treatment. In standard

electrochemotherapy treatment, after intravenous or intratumoral injection of chemotherapeutic delivered

electric pulses enhance the uptake of chemotherapeutics into tumor cells [17–19]. In electrochemotherapy,

chemotherapeutics cisplatin and bleomycin are commonly used as described in the Standard Operating

Procedures [18]. Starting in the late 1980s [20–22] with preclinical experiments, followed by clinical

studies [23,24], intratumoral injection of cisplatin was introduced into the Standard Operating Procedures of

the electrochemotherapy [18] and is now a well-established therapy in clinics across Europe [10]. Because

the mass of cisplatin can be determined relatively easy, we used cisplatin in our study. Cisplatin enters cells

by passive (diffusion) as well as active (endocytosis, pinocytosis, macrocytosis, membrane transporters)

mechanisms [25–27]. Tumor cells can become resistant to cisplatin due to several mechanisms; modified

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transport or detoxification, or increased DNA damage repair [27]. However, if the mechanism of resistance

depends on membrane restriction of cisplatin uptake or increased pumping out of the cell, the resistant and

non-resistant cells responded similarly to electroporation in vitro [28]. The situation in vivo was different,

as electroporation was more effective on parental than cisplatin-resistant tumors, and it can only be

hypothesized that additional factors could be present. Cisplatin is known to have negative side effects,

among them nephrotoxicity, ototoxicity, nausea, depending on the cisplatin concentration and

administration of the drug [29]. When injected, it accumulates in different organs (blood, liver, kidneys, and

brain) [30]. Thus, electrochemotherapy offers a possibility to deliver lower dose than in chemotherapy

while increasing local effectiveness. When treating patients with electrochemotherapy, we can follow

standard operating procedures where electrode geometry, parameters of electric pulses and

chemotherapeutic dose are predefined [18]. When tumors are outside the standard parameters, we can use

variable electrode configuration [31]. With variable electrode configuration, we need a treatment plan

where the position of the electrodes and electric pulse parameters are optimized to offer sufficient tumor

coverage with above-threshold electric field [32–36]. Current treatment planning procedure could be

upgraded by introducing a model of transport to the treatment plan. In this case, the spatially and temporally

dependent number of intracellular cisplatin molecules could be predicted. Number of bleomycin molecules

which must enter the cell to cause cell death has already been determined to be a few thousand [37]. If a

similar dependency, i.e., a necessary number of molecules for cell death, would also be revealed for

cisplatin, treatment efficacy could be more precisely predicted.

Usually, experiments are first performed in vitro and then translated to the in vivo environment. However, it

is difficult to compare the transport during electroporation in vivo and in vitro. In in vitro experiments with

single cell suspensions, the solute surrounds the cells, and the transport of solute occurs via diffusion.

Solute can pass cell membrane where the membrane is permeabilized. The transport in the extracellular

space is not limited. If there is enough of solute in the extracellular space, the transport stops when a cell

membrane reseals after electroporation. The transport through the membrane can be directly linked to the

cell membrane permeability [38]. In vivo, the initial solute concentration varies spatially. All the solute

available can enter cells before the membrane reseals and although cell membrane is still permeable, it does

not enter cells anymore, i.e., the extracellular compartment can be locally depleted. The transport of solute

occurs via diffusion and convection [39]. Cells are close together which decreases the possible area for the

uptake and decreases the induced transmembrane voltage due to electric field shielding [40]. Already in

spheroids, it was shown that transport of small molecules was spatially heterogeneous, cells inside the

spheroid contained less of solute than cells at the rim of the spheroid due to established concentration

gradient [39,41]. Electric pulses cause vasoconstriction [42] which limits the transport of solute in or out of

a tumor and prolongs the exposure of the tumor to chemotherapeutic. In a tumor, increased interstitial

pressure, heterogeneous perfusion, defective lymphatic circulation, binding of the drug to non-target

molecules and metabolism additionally affect the transport [43–45].

Modeling is advantageous since it decreases the number of necessary experiments and facilitates

understanding of the underlying processes. Several models of the increased uptake of molecules after

electroporation exist. For example, statistical models which describe the data well but do not include

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specific transport mechanisms [46], the kinetic scheme of electroporation [47,48], (electro)diffusion

through a permeable membrane or a single pore [49–53] and pharmacokinetic models [54,55]. We decided

to use the dual-porosity model [38,56] which is based on the diffusion equation, connects pore formation

with membrane permeability, and includes transport between several compartments – in our case between

intracellular space, interstitial tumor fraction and peritumoral environment, and can also include thermal

relations [57].

In our study, we focused on the electrochemotherapy and modeling of the drug transport after delivery of

electric pulses, i.e., electroporation. Our hypothesis was that in vitro membrane permeability coefficients

can be used to predict the transport of cisplatin in in vivo experiments. We used cisplatin as its intracellular

mass can be determined via measuring the mass of Pt by inductively coupled plasma mass spectrometry

[58] which is a very sensitive and precise method. We exposed suspension of mouse melanoma cells to

electric pulses in the presence of chemotherapeutic cisplatin and measured the intracellular mass of Pt. We

used the dual-porosity model to calculate the in vitro permeability coefficient as a function of pulse number

and electric field. We performed in vivo experiments on mouse melanoma tumors and measured the

intracellular and extracellular Pt mass, the mass of Pt in the serum, the mass of Pt bound to DNA and

components other than DNA (i.e., proteins and lipids). We built a 3D numerical model of a tumor and by

using the in vitro permeability coefficient predicted the mass of Pt in a tumor. A transformation from in

vitro to in vivo was possible by introducing a transformation coefficient which takes into account the

smaller available area of cell membrane for transport due to cell density, and presence of cell-matrix in

vivo. Our model can describe the amount of intracellular Pt and Pt in the interstitial fraction after

electroporation. We offer a step forward in connecting the transport at the cell level (in vitro) with the tissue

level (in vivo) which could be eventually used in treatment planning.

2. Materials and Methods

2.1. In vitro experiments

2.1.1 Cell preparation

Mouse skin melanoma cell line B16-F1 was grown until 80% confluency in an incubator (Kambič,

Slovenia) at 37°C and humidified 5% CO2 in Dulbecco's Modified Eagle’s Medium (DMEM, cat. no.

D5671, 13.77 mS/cm 317-351 mOs/kg). The cell line was tested to be mycobacterium free. DMEM was

supplemented with 10% fetal bovine serum (cat. no. F7524), 2 mM L-glutamine (cat. no. G7513) and

antibiotics (50 μg/mL gentamycin (cat. no. G1397), 1 U/mL penicillin-streptomycin (cat. no. P11-010,

PAA, Austria)) and was in this composition also used as electroporation buffer and for dilution of the

samples.

The cell suspension was prepared by detaching the cells by 10x trypsin-EDTA (cat. no T4174), diluted 1:9

in Hank’s basal salt solution (cat. no. H4641). Trypsin was inactivated with the DMEM containing 10%

fetal bovine serum. For electroporation, cells were centrifuged (5 min, 180g, 21°C) and resuspended in

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DMEM containing 10% fetal bovine serum at concentration 2.2x107 cells/mL. The chemicals were from

Sigma Aldrich, Germany, unless noted otherwise.

2.1.2. Cell electroporation

The 3.3 mM stock cisplatin (Accord Healthcare, Poland) was diluted in 0.9% NaCl and prepared fresh for

each experiment. Right before experiments, 120 μL of cell suspension was mixed with required quantity of

cisplatin to achieve the desired concentration (usually 100 μM, unless noted otherwise). 60 μL of the cell

suspension was transferred between the 2 mm stainless-steel electrodes [59], and pulses were delivered with

a commercially available BetaTech electroporator (Electro cell B10, BetaTech, France). We varied the

pulse number (1, 4, 8, 16, 32, 64) and the electric field (0.4 kV/cm, 0.6 kV/cm, 0.8 kV/cm, 1.0 kV/cm,

1.2 kV/cm) while pulse duration was 100 μs and repetition frequency 1 Hz. The remaining 60 μl was used

as a control and was transferred between the electrodes, but no pulses were delivered. 50 μl of the treated

and control sample was transferred to 15 mL centrifuge tubes. 10 min after pulse delivery (unless noted

otherwise), the samples were diluted 40x in full DMEM and vortexed. Samples were then centrifuged (5

min, 900g, 21 °C) and the supernatant was separated from the pellet. Until the Pt mass measurements the

samples were stored at -20 °C.

The following parameters were tested during establishement of our protocol: 1) the effect of extracellular

cisplatin (1 μM, 5 μM, 10 μM, 20 μM, 50 μM, 100 μM, 200 μM, 330 μM) on the intracellular Pt mass, 2)

time interval (5 min, 10 min or 20 min) between pulse delivery and dilution on the intracellular Pt mass, 3)

the binding of cisplatin to proteins from fetal bovine serum by using DMEM with or without fetal bovine

serum as the electroporation buffer, 4) we also determined the resealing time by adding 100 μM cisplatin 2

min, 5 min, 10 min and 20 min after pulse delivery. In the experiments mentioned in this paragraph,

8x100 μs pulses at 1.0 kV/cm and 1 Hz repetition frequency were delivered.

2.1.3. Cell death - Irreversible electroporation

Survival after electroporation (without cisplatin) was assessed with two different assays. Short-term

irreversible electroporation was measured 1h after the treatment on flow cytometer by the propidium iodide

uptake. The long-term irreversible electroporation was measured 24h after the treatment by a metabolic

assay. Cells were prepared the same way as described under Cell preparation. Electroporation was

performed as described in Cell electroporation section, but we added 0.9% NaCl instead of cisplatin.

2.2.3.1. Short-term irreversible electroporation

After application of electric pulses, cells were incubated at 37 °C for 30 min. During these 30 min cells

resealed. We added propidium iodide (Life Technologies, USA) in final concentration 136 μM. As

reversibly electroporated cells were already resealed, only irreversibly electroporated cells stained. After 5

min in the propidium iodide, we measured fluorescence on the flow cytometer (Attune NxT; Life

Technologies, USA). Cells were excited with a blue laser at 488 nm, the emitted fluorescence was detected

through a 574/26 nm band-pass filter, and 20.000 events were acquired. Two distinct peaks were formed on

the histogram of fluorescence and percentage of dead cells was determined with Attune NxT software (Life

Technologies, USA).

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2.2.3.2. Long-term irreversible electroporation

Long-term death was assessed 24h after electroporation with the MTS test (CellTiter 96® AQueous One

Solution Cell Proliferation Assay (MTS), Promega, USA). 20.000 cells per well were transferred into in a

96-well plate in triplicates and then incubated for 24 hr. 20 μL of the MTS was added per well, and after

2 hours at 37 °C, the absorbance at 490 nm was measured with a spectrofluorometer (Tecan Infinite 200;

Tecan, Austria).

2.2.4. Statistical Analysis

The statistical analysis was performed in SigmaPlot (v11.0, Systat Software, USA). We performed the t-test

when the normality test passed, or the Mann-Whitney rank sum test when the normality test failed.

2.2. In vivo experiments

2.2.1. Animals

Eight-week-old female C57Bl/6 mice (Envigo Laboratories, Udine, Italy) were used in experiments. At the

beginning of the experiments, their body weight was between 18 and 20 g. All procedures were performed

in compliance with the guidelines for animal experiments of the EU Directives, the permission of the

Ministry of Agriculture and the Environment of the Republic of Slovenia (Permission No. U34401-

1/2015/16), which was provided based on the approval of the National Ethics Committee for Experiments

on Laboratory Animals. Mice were kept in a specific pathogen-free environment with 12-hour light/dark

cycles at 20–24 °C with 55% ± 10% relative humidity and given food and water ad libitum.

2.2.2. Treatment and sample collection protocol

One day before the experiment, mice were shaved on their right flanks. Tumors were induced in the right

flanks of the non-anesthetized mice with subcutaneous inoculations of 106 B16-F10 melanoma cell

suspensions in 100 μL of a physiological solution prepared from cell culture in vitro. The treatment was

performed when tumors reached volumes of 35–40 mm3, which was 6-7 days after inoculation. Tumor

volume was measured using a Vernier Caliper. The tumor volume was calculated using the following

formula:

V=π×e1×e2×e3÷6; (1)

where e1, e2, and e3 are three orthogonal diameters of the tumor [60]. During the treatment, mice were under

isoflurane (Izofluran Torrex para 250 ml, Chiesi Slovenia, Ljubljana, Slovenia) anesthesia. To obtain

precise application of electroporation mice were initially anesthetized with inhalation anesthesia in the

induction chamber with 2% (v/v) of isoflurane in pure oxygen and afterward, the mouse muzzle was placed

under inhalation tube to keep mice anesthetized during the experiment. In experiments from [60], mice

were randomly divided into three experimental groups, consisting of three animals per group. The three

groups were: 1) control group, where we intratumorally injected 80 μL of physiological solution, 2)

cisplatin group, where we injected 80 μL of cisplatin (40 μg of cisplatin) and 3) cisplatin and

electroporation group, i.e. electrochemotherapy, where we injected 80 μL of cisplatin (40 μg of cisplatin)

and after 2 minutes delivered electric pulses. Pt mass in these three groups was determined 60 minutes after

injection as described in [60].

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In the scope of this paper, additional two groups were used with three mice per group. The treatment

consisted of an intratumoral injection of 80 μL cisplatin (40 µg of cisplatin). After two minutes, tumors

were excised, removed from the overlaying skin and weighted. The blood was collected from the

intraorbital sinus using glass capillary tubes. In the first group, we measured Pt in a whole tumor, and in the

second group, Pt in several tumor compartments (serum, single cell suspension, extracellular matrix, DNA)

[60]. For total tumor Pt content detection, tumors were frozen at -20 °C until the analysis.

2.3. Platinum measurements For Pt determination inductively coupled plasma mass spectrometry (ICP-MS) was used.

2.3.1. Sample digestion

All dilutions of the samples were made with ultrapure water (18.2 MΩ cm) obtained from a Direct-Q 5

Ultrapure water system (Merck Millipore, USA). Nitric acid (65 % HNO3) and hydrogen peroxide (30 %

H2O2) were obtained by Merck Millipore, USA.

Samples of cell pellet were digested with 200 µL of concentrated nitric acid, 200 µL of concentrated

hydrogen peroxide and heated overnight at 90°C in the heating oven (Binder GmbH, Tuttlingen, Germany).

After the digestion, the samples were filled up to 5 mL with MilliQ water and measured by ICP-MS.

Samples of supernatant fluid were digested with one mL of concentrated nitric acid, one mL of

concentrated hydrogen peroxide and heated overnight at 90°C in the heating oven. After the digestion, the

samples were filled up to 10 mL with MilliQ water and 2.5-times diluted before ICP-MS measurements.

Samples from the in vivo experiments were digested at 90 °C for approximately 36 h in 0.2 to 2.0 mL of a

mixture of concentrated nitric acid and hydrogen peroxide (1:1). After obtaining clear solutions, samples

were adequately diluted with MilliQ water before ICP-MS measurements.

2.3.2. Inductively coupled plasma – mass spectrometry measurements

Mass concentrations of Pt in the digested samples were determined by the use of mass spectrometry with

inductively coupled plasma (ICP-MS) against an external calibration curve using Ir as an internal standard.

Calibration standard solutions of Pt were prepared from Pt stock solution (1000 µg Pt/mL in 8 % HCl),

obtained from Merck (Darmstadt, Germany), and diluted in 2.6 % nitric acid. Experimental conditions for

the ICP-MS instrument (Agilent 7900 ICP-MS instrument, Agilent Technologies, Tokyo, Japan),

summarized in Table 1, were optimized for plasma robustness and adequate sensitivity.

For evaluation of the accuracy of the analytical method (digestion procedure and ICP-MS analysis), the

same quantity of cell pellet was spiked with 20 µL of 500 ng/mL ionic Pt solution to reach final Pt

concentration of 10 ng/sample, while the supernatant fluid was spiked with 20 µL of 50 µg/mL ionic Pt

solution to reach final Pt concentration of 1000 ng/sample. Spiked samples were digested in the same way

as described before. Recoveries (the ratio between the measured and expected Pt concentrations) were 101

± 2 % (N=4) and 98 ± 1 % (N=4) for spiked cell pellets and spiked supernatant fluids, respectively.

Parameter Type/Value

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Sample introduction

Nebulizer Micromist

Spray chamber Scott

Skimmer and sampler cone Ni

Plasma condition

Forward power 1550 W

Plasma gas flow 15.0 L min-1

Carrier gas flow 1.05 L min-1

Nebulizer pump 0.1 rps

Sample depth 8.0 mm

He gas flow 0 mL min-1

Energy discrimination 5.0 V

Data acquisition parameters

Isotopes monitored 195Pt

Isotopes of internal standards 193Ir

2.4. Modelling

2.4.1. In vitro

From the in vitro results, we calculated the coefficient membrane permeability to Pt (P). In vitro, the model

of transport was relatively simple – the initial concentration of the extracellular cisplatin was the same for

all the cells, there was enough of cisplatin so that we could consider the extracellular cisplatin concentration

constant. The whole membrane was surrounded by cisplatin which could enter from all sides, depending on

the permeability of the membrane. By changing the pulse number and voltage, we obtained the permeability

coefficient as a function of pulse number and electric field intensity. Permeability coefficient was a direct

measure of membrane permeability.

The resealing time constant was determined by describing the resealing data with a curve:

𝑓𝑓(𝑡𝑡) = 𝐶𝐶 exp �− 𝑡𝑡𝜏𝜏� (2)

Where C denotes the intracellular Pt mass if cisplatin is present at the moment of pulse delivery, τ is the

resealing time constant or time when 63% of the pores/defects are resealed [61], and t is the time elapsed

since the end of pulse delivery.

In vitro, we used the dual-porosity model [38] to calculate the permeability coefficient. In Matlab (R2017a,

Mathworks Inc., USA) we calculated the permeability coefficient (P) at t = 0 s which corresponds to P0 by

solving the differential equation:

𝜕𝜕𝑐𝑐𝑖𝑖𝑖𝑖𝑖𝑖𝜕𝜕𝑡𝑡

= 𝑃𝑃0exp (− 𝑡𝑡𝜏𝜏 ) 3𝑅𝑅

(𝑐𝑐𝑖𝑖𝑛𝑛𝑡𝑡 − 𝑐𝑐𝑒𝑒𝑒𝑒𝑡𝑡) (3)

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for the external concentration cext (100 μM), the resealing constant τ (2.29 min), in vitro cell radius R

(8.1 μm) [62], and initial conditions cint(t=0s)=0 μM. The permeability coefficient was a function of time as

the permeability decreased with time due to membrane resealing.

2.4.2. In vivo

Model of a tumor was constructed in Comsol Multiphysics® (v5.3, Comsol AB, Stockholm, Sweden). The

tumor was modeled as a sphere with radius 2.03 mm which corresponds to 35 mm3, i.e., the average volume

of a tumor when in vivo treatment was performed. Due to symmetry, we modeled only 1/8 of the tumor,

i.e., one octant of the sphere. We used physics-controlled mesh with extra fine element size and confirmed

that with decreasing element size, the calculated intra- and extracellular Pt mass did not change

significantly. In Comsol, two equations were coupled and calculated in a whole tumor. In the intracellular

space:

𝜕𝜕𝑐𝑐𝑖𝑖𝑖𝑖𝑖𝑖𝜕𝜕𝑡𝑡

= −3𝑃𝑃(𝑡𝑡)𝑅𝑅

(𝑐𝑐𝑖𝑖𝑛𝑛𝑡𝑡 − 𝑐𝑐𝑒𝑒𝑒𝑒𝑡𝑡)𝑘𝑘 (4)

And in the extracellular space:

𝜕𝜕𝑐𝑐𝑒𝑒𝑒𝑒𝑖𝑖𝜕𝜕𝑡𝑡

= 𝐷𝐷𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐∇2𝑐𝑐𝑒𝑒𝑒𝑒𝑡𝑡 + 3𝑃𝑃(𝑡𝑡)𝑅𝑅

(1−𝑓𝑓)𝑓𝑓

(𝑐𝑐𝑖𝑖𝑛𝑛𝑡𝑡 − 𝑐𝑐𝑒𝑒𝑒𝑒𝑡𝑡)𝑘𝑘 (5)

Where cint and cext denote the intracellular and extracellular concentration, respectively, Dcddp is the

diffusion coefficient of cisplatin in the extracellular tumor space, P is the permeability coefficient, time-

dependent due to membrane resealing, R is the cell radius in vivo (6.25 μm) [63], f is the fraction of cells in

a tumor (37%) [63,64], and k is the transformation coefficient which transforms the data from the in vitro to

the in vivo conditions. The initial conditions were cint(t=0)=0 mM and cext(t=0)=1.6 mM. P was determined

from the in vitro experiments. The boundary conditions for the intracellular concentration were no flux on

all boundaries and for extracellular concentration, no flux in the inner boundaries of a tumor and cext(r=2.03

mm)=0 mM at external boundaries.

In a model, we varied parameters k and Dcddp and calculated the intracellular and extracellular Pt mass for

each combination of k and Dcddp when k was between 0 and 1 in steps of 0.2 and Dcddp was between 2 x10-8

cm2/s and 8x10-5 cm2/s in steps of 0.2 decades. Then, we determined a global minimum and sampled the

space around it with higher resolution by changing k from 0.3 to 0.5 by 0.1 and Dcddp from 1.5x10-6 cm2/s to

2.5 x10-6 cm2/s by 0.1 decades. The final values of the parameters k and Dcddp were chosen as the values

with which we achieved the best agreement with a measured mass of Pt in vivo and were the global

minimum.

3. Results

3.1. In vitro experiments First, we performed experiments to determine the effect of extracellular cisplatin concentration, incubation

time, fetal bovine serum, and the resealing time on the Pt uptake. From Fig. 1 we can see that the

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intracellular Pt mass was linearly dependent on the extracellular cisplatin concentration in control as well as

in the electroporated cells. Thus, we decided to perform all subsequent experiments with 100 μM

extracellular cisplatin and scaled our results to any concentration in the tested range. In Fig. 2 we can see

that 5 min after pulse delivery we reach a plateau in the intracellular Pt mass. We thus decided to perform

all subsequent experiments with 10 min incubation time after electroporation. Next, we tested whether the

presence of serum in the electroporation medium affects cellular uptake of Pt (Fig 3). There was no

statistical difference between the uptake with or without serum. In Fig. 4 we can see how the membranes

resealed after pulse application. When describing the data with a first-order dynamics (Eq. (1)), we

determined that the resealing time constant was 2.29 min. 10 min after pulse delivery was cell membrane

resealed which corroborates results shown in Fig. 2.

Fig. 1: Effect of extracellular cisplatin (Cext) concentration on intracellular Pt mass when 8x100 μs

pulses of 1 kV/cm were delivered at repetition frequency 1 Hz. The results could be described with a linear

function y = kcextwith k and R2 of 0.2672 and 0.99 for the treated and 0.0228 and 0.92 for the control

cells, respectively. Mean ± standard deviation. Each data point was repeated 3-5 times. Except for 1 μM

extracellular cisplatin where the measured values were below the detection threshold of 0.005 ng and we

could not evaluate statistical significance, all electroporated samples were statistically different from the

corresponding control samples (P<=0.03).

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Fig. 2: Effect of incubation time after pulse application on the intracellular Pt mass when 8x100 μs

pulses of 1 kV/cm were delivered at repetition frequency 1 Hz and 100 μM extracellular cisplatin. Already 5

min after pulse delivery a plateau was reached. Mean ± standard deviation. Each data point was repeated

3-6 times. All treated samples were statistically different from the corresponding control (P<=0.001).

There was a significant difference between 2 min and 5 min exposure of treated samples (P=0.02), other

comparisons were not significant.

Fig. 3: Effect of the presence of serum in the electroporation medium on intracellular Pt mass. There

was no statistical difference in pulsed (P=0.652) and control (P=0.430) cells when serum was or was not

present (t-test). Mean ± standard deviation. Each data point was repeated 3-5 times.

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Fig. 4: Membrane resealing after 8x100 μs pulses of 1 kV/cm were delivered at repetition frequency 1 Hz

and 100 μM cisplatin (mean ± standard deviation). Membrane resealed with a first-order dynamics with a

time constant 2.29 min which is shown in red. Each data point was repeated 4-6 times. 2 and 5 min data

point was statistically significantly different between electroporated and corresponding control samples

(P<=0.001 for 2 min and P=0.045 for 5 min). There was also a statistical difference between 2 min and 5

min electroporated samples (P=0.002). All other comparisons were not significant.

In Fig. 5 intracellular Pt mass is shown as a function of pulse number and electric field intensity. We can

see that the highest uptake was achieved with eight pulses of 1.2 kV/cm (35 ng of Pt per sample). With

higher pulse number or higher electric field, the intracellular Pt mass decreased, most likely due to

increased cell death. To explore hypothesis of cell death contribution, we also performed experiments

evaluating cell death. Results are shown in Fig. 6 - in Fig. 6a is the short-term cell death measured by flow

cytometry one hour after the treatment and in Fig. 6b is the metabolic activity measured by MTS assay 24

hours after the treatment. We can see that the results obtained with both assays correlate well. Survival

starts to decrease at electric fields above 0.6 kV/cm for any pulse number tested. With 1, 4 or 8 pulses even

at the highest electric fields, the survival was not yet affected. For calculating the permeability coefficients,

we have to take into account cell death and correct the permeability coefficient.

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Fig. 5: Effect of number of pulses and the electric field on the intracellular Pt mass. 8x100 μs pulses of 1

kV/cm were delivered at repetition frequency 1 Hz (mean ± standard deviation). Each data point was

repeated 3-5 times. Data points were compared to the control. Comparison was not significant for 0.4

kV/cm and 1, 4 or 32 pulses; for 0.6 kV/cm and 1, 4, 8 or 64 pulses, for 0.8 kV/cm and 1 or 4 pulses. All

other data points were statistically significant (P<=0.05).

Fig. 6: Decrease in cell survival due to irreversible electroporation. (a) Survival determined by flow

cytometry, one hour after electroporation and (b) by the MTS survival assay, 24 h after electroporation

(mean ± standard deviation). The legend is valid for both figures. Each data point was repeated 3-6 times.

In (a) the statistically significant points in comparison to the control were 0.8 kV/cm and 32 or 64 pulses, 1

kV/cm at 8, 16, 32 or 64 pulses and 1.2 kV/cm at 16, 32 or 64 pulses (P<=0.03). In (b) the statistically

significant points in comparison to the control were 0.4 kV/cm at 8, 32 and 64 pulses, 0.8 kV/cm at 64

pulses, 1 kV/cm at 1, 32 or 64 pulses, 1.2 kV/cm at 8, 32 and 64 pulses (P<=0.03).

Due to increased cell membrane permeability, different molecules and ions can enter the cell in different

concentrations, depending on their size, molecular weight and charge. Small molecules (several hundredths

of Da) were shown to enter cells in 90-100% of their extracellular concentration [58]. In our experiments,

we calculated intracellular cisplatin concentration in comparison to extracellular cisplatin concentration

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(Fig. 7). We reached 80% of the extracellular concentration when 8 pulses of 1.2 kV/cm were applied (Fig.

7a). Lower cell membrane permeability or cell death decreased this percentage in case of other pulse

parameters. We took cell death into account by normalizing the intracellular concentration to the survival as

determined by flow cytometry and obtained graph in Fig. 7b. There we can see that we obtain uptake higher

than 100% and is increasing with increasing electric field and pulse number.

Fig. 7: The fraction of the extracellular Pt concentration, present in the intracellular compartment

(mean ± standard deviation). a) Cell death was not taken into account, and the intracellular Pt

concentration was almost 80% of the extracellular cisplatin concentration. b) Results of the uptake were

corrected for cell death as determined by flow cytometry one hour after the treatment. The intracellular Pt

concentration was almost 140% of the extracellular cisplatin concentration. The legend is valid for both

figures.

3.2. In vivo experiments We measured the intra- and extracellular Pt mass and by taking into account the size of cells and their

volume fraction also calculated the intracellular and extracellular cisplatin concentration. The concentration

is just an approximation since the concentration in the tissue/tumor is most likely inhomogeneous. Results

are shown in Table 2.

Table 2: Intra- and extracellular Pt mass and the corresponding cisplatin concentration, based on the

measured volume of each tumor at t=2 min after injection of cisplatin (initial conditions, time when electric

pulses are applied) and at t=60 min after the therapy (cisplatin + electroporation). From the results at 60 min

we subtracted the Pt in the control samples (without electroporation) to focus only on the transport due to

electroporation.

t = 0 min

(Mean ± St. dev).

t = 60 min [60]

(Mean ± St. dev.)

Intracellular Pt mass 17 ng ± 12 ng 67 ng ± 18 ng

Extracellular Pt mass 6431 ng ± 4681 ng 666 ng ± 104 ng

Intracellular cisplatin concentration 7 μM ± 5.5 μM 25 μM ± 10 μM

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Extracellular cisplatin concentration 1600 μM ± 1200 μM 144 μM ±110 μM

3.3. Modeling By using the dual-porosity model, we calculated the in vitro permeability coefficient. We also took into

account that the permeability coefficient changes with time due to cell membrane resealing. We assumed

that during resealing, the permeability of the plasma membrane decreases exponentially with time, which

has been confirmed experimentally and by modeling in several studies [61,66,67]. Thus, the permeability

coefficient also decreases exponentially with time. The resealing coefficient was obtained by fitting Eq. (2)

to results, shown in Fig. 4 and optimizing the τ to obtain the highest R2 value of the model. In Fig. 8a we

can see the permeability coefficient P0 at t=0 s, i.e., when electric pulses were applied and the cell

membrane was fully permeabilized and in Fig. 8b, how the permeability coefficient changed with time.

Fig. 8: The permeability coefficient at pulse application determined by the dual-porosity model [32] for

all conditions, where survival was 100%. a) Permeability coefficient at t=0, i.e., just after exposure to

electric pulses. b) Permeability coefficient as a function of time when 8x100 μs pulses were delivered at

0.4 kV/cm and 1 Hz repetition frequency on a semi-logarithmic scale. This was the permeability coefficient

which was used in the in vivo model.

For the in vivo conditions, we modeled transport in a tumor modeled by a sphere consisting of intra- (tumor

cells) and extracellular (interstitial fraction) space. In each compartment, the transport was described by the

dual-porosity model. We determined the optimal diffusion coefficient to be 2.1x10-6 cm2/s and the optimal

transformation coefficient to be 0.3. We obtained temporal and spatial dynamics of Pt uptake in the intra-

and extracellular environment. To compare results of the simulation with experimental results [60], we

extracted from the model results one hour after pulse delivery. In the model, the Pt mass achieved plateau

sooner since cell membrane resealed in 10 min (Fig. 4) but we simulated one hour more to be in the same

time point as when tumors were excised.

The concentration of Pt in a tumor one hour after electroporation as calculated by the model was

inhomogeneous with higher concentration in the center of a tumor and gradually decreasing to the rim of a

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tumor, where concentration was 0 mM (Fig. 9). Therefore, we compared the mass of Pt which was

calculated as an integral over the corresponding volume fraction to the measured Pt mass in the in vivo

experiments.

Fig 9: Modelled spatial dependency of Pt concentration 1 hour after pulse delivery in mol/m3. (a) Intra-

and (b) extracellular Pt concentration. Spatial distribution is inhomogeneous in both cases.

Table 3: Experimentally determined and modelled mass of Pt in the intracellular and extracellular space of

a tumor 1 hour after the treatment with electrochemotherapy [54].

Experiments [60] Model

Intracellular Pt mass 67 ng 65 ng

Extracellular Pt mass 666 ng 664 ng

4. Discussion Our aim was to model the transport of chemotherapeutics in tumors based on experiments obtained on cell

suspension. On cell suspension, experiments are easier to perform, there are no ethical issues, and a large

spectrum of parameters can be explored. Our goal was to include a model of transport in treatment planning

of electroporation-based medical treatments. In treatment planning of electrochemotherapy, a critical

experimentally determined electric field of permeabilization [68,69] or statistical models [59,70] are used to

determine which area will be permeabilized/affected and which not. Permeabilization alone, however, does

not guarantee treatment efficacy [10]. For successful electrochemotherapy, a sufficient amount of

chemotherapeutics has to enter tumor cells and bind to DNA (the main target of cisplatin) or target other

molecules in the cell to prevent future cell division and cause cell death. Current treatment plans of

electrochemotherapy lack a model of transport and drug uptake in a tumor and the cells. A number of

internalized bleomycin has been correlated with the biological effect – few thousand bleomycin molecules

cause mitotic death while several million molecules cause cell death similar to apoptosis [37,71]. A similar

analysis could also be done for cisplatin.

In our paper, we connected the in vitro experiments with the in vivo experiments by the dual-porosity

model. We accurately calculated the Pt mass in the intra- and extracellular tumor space by using in vitro

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permeability coefficients. Our model could be used to predict treatment efficacy from a number of

molecules delivered to cells as a function of the electric field in tissue.

4.1. Experimental part In vitro, we used mouse melanoma B16-F1 cells and in vivo, tumor from mouse melanoma B16-F10 cells.

B16-F1 and B16-F10 cells vary only in their metastatic potential. [72] In vitro, increased uptake of

molecules after to electroporation is relatively easy to determine using fluorescent dyes and techniques like

fluorescent microscopy, flow cytometry, spectrofluorometry [73]. In vivo, the increased uptake of

molecules after electroporation is challenging to assess. 57Co-bleomycin [74], 111In-bleomycin [75], 99mTc-

DTPA [76], 51Cr-EDTA [77], gadolinium [78], lucifer yellow [79] and cisplatin [20,21,28,80–84] were

used previously. Cisplatin is Pt-based chemotherapeutic. We can calculate the concentration of cisplatin by

measuring Pt mass in a known volume. The mass of Pt can be measured precisely even for very low

amounts by inductively coupled plasma mass spectrometry. As Pt mass is easily determined in vitro as well

as in vivo, we used cisplatin in our study. Another possibility for Pt detection is attaching a contrasting

agent as a ligand to the Pt.

Before exploring the parameter space by changing the number of applied electric pulses and electric field,

we tested 1) at what extracellular cisplatin concentration we should perform the experiments, 2) if we could

use the fetal bovine serum, 3) how long should we incubate cells after pulse delivery and 4) what was the

effect of membrane resealing on intracellular Pt mass. 1) We determined that with increasing extracellular

cisplatin concentration, also intracellular Pt mass increased. The dependency was linear which indicates that

the increased uptake of molecules after electroporation was mostly diffusive and not electrophoretic or

endocytic which is in accordance with an already published study [85]. We decided to perform experiments

at 100 μM and because of the linear dependency results could be scaled to any concentration in the tested

range. 100 μM concentration was used because it is in a similar range as in other in vitro studies [22,86–88]

and higher cisplatin concentration caused higher intracellular Pt mass which enabled to more accurately

distinguish between intracellular masses at similar pulse parameters. 2) Fetal bovine serum added to the

growth medium consists mostly of proteins. Cisplatin unspecifically binds to proteins [58] and also to

collagen [89] which represents 5-10% of the matrix. The time dynamics of cisplatin binding to proteins is in

a range of hours [58,90,91]. Cisplatin, bound to proteins, is biologically inactive [92]. In vitro the binding

of cisplatin to proteins in the fetal bovine serum in the first 10 minutes was negligible (Fig. 3). Because

cisplatin was observed to bind to proteins in a time-range of hours-days, we did not include cisplatin

binding in the tumor model. 3) When we tested the influence of incubation time after electric pulse

application, we determined that a plateau is reached after 5 minutes (Fig. 2). Thus we decided to perform all

our experiments 10 min after pulse application when the uptake due to electroporation ends. The reason for

reaching a plateau could be that the increased uptake of molecules after electroporation is much faster than

the passive diffusion or active mechanisms which exist in the cells. Namely, it was shown that without

electroporation, intracellular cisplatin concentration increases in a time-range of hours, in vitro as well as in

vivo [30,93] The additional mechanisms could contribute to a higher transport and increase in the uptake on

a longer timescale. To focus only on the modeling uptake due to electroporation, we subtracted the mass of

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the Pt in the control samples from the mass of Pt in treated samples when calculating the permeability

coefficient and comparing the results of our model with the experiments.

Electrochemotherapy is usually performed with 8x100 μs pulses. However, we wanted our model to be

valid in a larger parameter space, which would enable its use also with other pulse parameters. Thus, we

applied from 1 to 64 pulses. With increasing pulse number and electric field the intracellular Pt mass

increased with a peak at eight 100 μs pulses of 1.2 kV/cm (Fig. 5). We calculated the volume of all cells (≈2

mm3) in a sample and using the measured Pt mass in the cells calculated the intracellular Pt concentration.

We determined that the intracellular Pt concentration is up to 80% of the extracellular cisplatin

concentration (Fig. 7a). The high percentage is in agreement with the mass of the cisplatin (300 g/mol) [65].

With some parameters of electric pulses, we were already irreversibly electroporating cells (Fig. 6) which

also affected the uptake. The uptake was decreased as can be observed in the shape of the curves in Fig. 5,

i.e., we reach maximum at lower pulse number (eight pulses at 1.2 kV/cm or 16 pulses at 1.0 kV/cm) and

then with increasing pulse number the uptake decreases. We normalized the percentage of intracellular

cisplatin concentration to survival and obtained values higher than 100% (Fig. 7b) which indicates that

either 1) cisplatin binds to DNA and concentration gradient increases, 2) cell death was not the only reason

for decreased uptake, 3) some other cell death assay would be more suitable, 4) active mechanisms

additionally transport cisplatin into cells. We determined that eight pulses are the optimal number to be

delivered which is consistent with the standard electrochemotherapy protocol [18]. For our in vivo model,

we only used the permeability coefficient when the survival was 100%.

In the in vivo experiments, we injected 26 μg of Pt by intratumoral injection of 80 μl of cisplatin in 35 mm3

tumors. The volume of injection was twice the volume of a tumor because we reused the data from [60] to

follow the 3Rs of the animal experiments [94]. Most of the injected Pt was not measured. When a tumor

was excised 2 min after cisplatin injection, we determined that 27 % of the injected Pt was still present in a

tumor and the serum. A part of cisplatin was lost due to lower tumor volume than injection volume and due

to washout of cisplatin from the tumor. When a tumor was excised after 1 hour, we determined that in a

tumor and serum there was 2% of the injected Pt in the control samples (cisplatin injection, no

electroporation) and 7 % in the treated (cisplatin injection and electroporation) samples. Thus, in vivo, the

majority of cisplatin was unaccounted for. Part of cisplatin was lost due to the used methodology (when the

tumor was mechanically disintegrated and during blood collection). Part of cisplatin could accumulate in

different organs and cause unwanted side effects [95].

4.2. Modelling part Our model is based on a hypothesis that cell permeability in vitro and in vivo is similar when cells are

exposed to the same induced transmembrane voltage. However, the transport in vivo is decreased in

comparison to the transport in vitro due to the hindered transport of molecules in the in vivo environment

due to the cell matrix, the lower diffusion coefficient of molecules in the interstitial space and close cell

contacts. The transformation of the transport model from the in vitro to the in vivo environment is

challenging. We can use the in vitro permeability coefficient in the in vivo model, as long as we take into

account the geometric characteristics of tissue (cell volume fraction, cell matrix volume fraction, cell size).

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We took the characteristics into account by decreasing the transport by the transformation coefficient. In

our model, we assume that the transformation coefficient is a function of tissue morphology and decreases

with increasing cell volume fraction and matrix volume fraction. It remains to be tested if the

transformation coefficient can be used over several values of parameters and if transformation coefficient is

a function of experimental conditions (e.g., tumor cell volume fraction) [63].

Our initial conditions of the in vivo model were homogeneously distributed 1.6 mM cisplatin in the tumor

interstitial fraction. Initial cisplatin concentration was determined from experiments by excising tumors 2

min after cisplatin injection when usually electric pulses were applied. According to our calculations, the

initial cisplatin concentration in a tumor (1.6 mM) was similar to the concentration injected (1.65 mM),

which means that by injecting cisplatin we washed out the interstitial fluid and replaced it with cisplatin.

Thus, part of cisplatin volume was lost due to washout from the tumor, but the remaining cisplatin was not

diluted in the interstitial fraction, and it remained in (almost) initial concentration. Initial cisplatin

distribution was determined from the literature [21,96]. Solutes are washed out relatively quickly from

small tumors and with some delay from larger tumors [96]. In [21] it was determined that

electrochemotherapy treatment was most efficient when electric pulses were delivered 0-5 min after

intratumoral cisplatin injection. This time scale indicates that in the first 5 minutes after intratumoral

injection the cisplatin distribution in a tumor is approximately homogeneous and not yet washed out of a

tumor. 0-5 minutes was a similar time range as used in our experiments (2 min) implying that

approximating the initial cisplatin distribution in the model as homogeneous is justified. In future, for

experimental determination of initial cisplatin distribution, a technique like imaging mass cytometry could

be used [89]. 60 minutes after the treatment, the cisplatin concentration was the highest in the center of the

tumor, which corresponds to previously published data – in [79] a fluorescent dye was injected, and in the

tissue sections it can be seen that the highest concentration of the dye was in the middle of the sample.

We modeled the tumor as a symmetrical sphere, thus decreasing the complexity of the model and time of

calculation, which was justifiable because mouse melanomas used in our experiments were spherical due to

a localized injection during inoculation. However, our model also supports also other geometries, the only

adaptations required would be those of the geometry and the mesh.

The volume fraction of cells in a tumor was calculated by the method presented in [64] - the surface

fraction of cells determined on a slice is a good approximation of volume fraction. We used experimental

data [63] where the volume fraction of cells and cell matrix and cell size in B16-F1 mouse melanoma

tumors were assessed. We calculated cell volume fraction of 37% in tumors. Since a tumor has a high

cellular fraction in comparison to our in vitro results, we took that into account. The high volume fraction of

cells decreased the induced transmembrane voltage due to electric field shielding [40]. Considering the

calculations in [97,98] we can determine that the induced transmembrane voltage in cells in vivo is 76-88%

of the one induced on a single spherical cell. The average radius of cells in vivo was determined to be 6.25

μm [63] which is less than the average in vitro radius of 8.1±1.1 μm [62] which means that according to

Schwann equation, the induced transmembrane on the cells in vivo is 77% of the in vitro induced

transmembrane voltage. Taking into account the decrease in induced transmembrane voltage due to cell

proximity and smaller cells in vivo, under the same external electric field, the induced transmembrane

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voltage is approximately 60% of the one induced on cells in vitro. The voltage-to-distance ratio delivered to

a tumor by parallel plate electrodes was 1.3 kV/cm. It was numerically determined that the electric field in a

tumor was inhomogeneous [36], and in the center of a tumor, it was 0.6 kV/cm. Taking into account the

lower induced transmembrane voltage on the cells in vivo, the conditions in vivo correspond to when 0.34-

0.4 kV/cm is delivered in vitro. Thus, the in vitro permeability coefficient when eight 100 μs, 0.4 kV/cm at

1 Hz was used in the tumor model. We also took into account cell resealing after pulse application and the

decrease in cell permeability coefficient. From the in vitro results, we determined the time constant of

resealing was 2.29 min which is in agreement with the in vivo data (Fig. 2 in [21]). When eight pulses at 0.4

kV/cm were delivered the survival was still 100% (Fig. 6) and thus there was no need for permeability

coefficient correction. The calculated value of 0.4 kV/cm is in agreement with the literature where 0.4

kV/cm is usually assumed to be a threshold of electroporation for a tumor [79,99].

In tumors, the transport differs from normal tissues, as tumors have an abnormal vasculature and increased

interstitial pressure which facilitates cisplatin loss [100,101]. In the interstitial space, two mechanisms of

transport are present – diffusion and convection. The diffusion in tissues is lower than in water as tissue

structures obstruct the transport. In the literature, values of diffusion coefficient for cisplatin or similar

molecules are reported in the range of 0.1-0.7x10-6 cm2/s [102–104] which is less than the value we

determined in our model (2.1x10-6 cm2/s). Electric pulses cause a vascular lock which is present in the time

range of hours, although blood flow is to some extent restored in first 15 minutes [42]. It is possible that the

diffusion constant in our study is higher because a part of the transport out of a tumor was due to

convection, while we limited our analysis to diffusion. However, convection introduces a new unknown

parameter (velocity) in the model, and here we decided to model the transport as only diffusive. Also,

vasoconstriction due to electric pulses decreases convective transport from the tumor. In future, the

convective transport could be included, but experimental data are first needed.

All models include simplifications. The following are simplifications we made in our model. We assumed

that the electric field in a tumor was homogeneous although in reality, it is inhomogeneous [36,79]. We

approximated initial cisplatin distribution in the tumor as homogeneous and the cisplatin concentration at

the external border of a tumor to be 0 mM, although in reality there is a smooth transition in the

concentration between intra- and extratumoral cisplatin. Our model was tested on one type of a tumor, and

more in vivo experiments are needed, preferably with several different tumor types to increase the validity

of the model. Our current hypothesis is that different tumor types will differ in the diffusion coefficient for

cisplatin and the transformation coefficient in the interstitial tumor fraction depending on the volume

fraction of cells in the tumor, both decreasing with the increased volume fraction of cells in the tumor.

Additionally, different sizes and volume fractions of cells in different tumors affect the induced

transmembrane voltage [97,98]. Permeability coefficient for different tumor types should be calculated

from the experiments on corresponding cell types in vitro as different cells exhibit different sensitivity to

electric pulses [105]. We have applied electric pulses in vivo at only one pulse number and electric field.

Thus our model is valid at 1.3 kV/cm voltage over the distance, 8x100 μs pulses between the electrodes and

should in future be tested with more parameters in vivo. Our model is valid for transport of small molecules

in the range of several hundred Da. For gene electrotherapy where large plasmids have to pass the cell

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membrane, a different approach would need to be used as the mechanisms governing the transport of large

molecules [106] are different from mechanisms governing the transport of small molecules [50,54,85].

Also, the mobility of large molecules such as plasmids in the tissue is smaller than the mobility of small

molecules like cisplatin [43]. We assumed that the resealing time in vitro and in vivo is similar, as was

suggested by comparing our in vitro data to the in vivo already published study [21]. The difference in the

resealing would influence the time dependency of the permeability coefficient. The size of the cells in vitro

is statistically distributed [54] which could in future be included in the calculation of the permeability

coefficient.

There are several possible upgrades to our model. We could include kinetics of the binding of cisplatin to

the DNA and resulting cell death as was previously described at a cell level [107–110]. We modeled

intratumoral cisplatin delivery as suggested in Standard Operating procedures of electrochemotherapy [18].

However, for our model to be useful for intravenous (i.v.) drug delivery, additional steps would have to be

performed to determine the initial spatial drug concentration in the tumor. In the i.v. delivery the drug has to

overcome several obstacles before entering the tumor – it has to extravasate from the blood vessels, reach

the tumor, overcome the increased interstitial pressure and distribute throughout the interstitial tumor

fraction [100]. With our model, we could predict not only the intracellular cisplatin concentration but also

cell death as a function of the electric field in the tissue and injected cisplatin dose. We would need to

additionally determine the sufficient amount of molecules in the cell to achieve cell death, and then predict

the apoptotic or necrotic area of the tumor.

5. Conclusion Electrochemotherapy is a combination of delivering chemotherapeutics and electric pulses to efficiently

treat tumors. To predict the treatment outcome or optimize the position of the electrodes, we first calculate

the electric field distribution. We use the critical electric field for reversible electroporation to determine

reversibly electroporated area, i.e., the area where electrochemotherapy would be efficient. In the paper, we

introduced a model of drug transport that described the transport of cisplatin after intratumoral injection and

application of electric pulses in subcutaneous melanoma tumor. We modeled the transport of

chemotherapeutic between several compartments – tumor cells, interstitial tumor fraction, and peritumoral

environment using the dual-porosity model. We described the uptake of cisplatin in vivo by using the

permeability coefficient from in vitro experiments. By optimizing the parameters of the treatment in vitro, a

large spectrum of parameters could be easily tested, and a number of experiments on animals could be

reduced. We offer a step forward in bridging the models at the cell level (in vitro) and the models at the

tissue level (in vivo). By incorporating the analysis of transport into treatment planning of

electrochemotherapy, the precision of the outcome could be increased taking into account the cell-level

mechanisms of electroporation.

Acknowledgements

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This work was supported by the Slovenian Research Agency (ARRS) [research core funding No. P2-0249, P3-0003,

and IP-0510, and funding for Junior Researchers to JDČ and KU]. The research was conducted in the scope of the

electroporation in Biology and Medicine (EBAM) European Associated Laboratory (LEA). JDČ would like to thank D.

Hodžić and L. Vukanović for their help in the cell culture laboratory, Dr. S. Mahnič-Kalamiza for discussions on the

dual-porosity model, and Dr. T. Kotnik for proofreading the paper.

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Conclusions and future work

Conclusions and future work In the scope of this doctoral dissertation, we have modeled electroporation at the level of single cells and

the level of tissues. At the level of single cells, experiments are relatively easy to perform, and a large

spectrum of parameters can be explored without ethical issues which are present in animal experiments

(Workman et al. 2010). However, not many models include single-cell properties in the bulk tissue models.

Our aim was to use results of cell level in vitro experiments to establish models which can be then used to

model electroporation in vivo.

First, we have employed mathematical models of cell membrane permeability and cell death to describe

results of in vitro experiments on cell suspensions and cell monolayers. We concluded that several models

describe the experimental data well. Our optimized Peleg-Fermi model of cell death was already used on

tissues with good accuracy of prediction (Kranjc et al. 2017). In the survival models, the models approach

survival zero but never reach it. It remains to be established how low the survival has to be to achieve a

complete response. The last few remaining cells in electrochemotherapy as well as in ablation with

irreversible electroporation are shown to be eradicated by the immune system (Neal et al. 2013; Serša et al.

2015).

We analyzed the time-dynamics of the dye uptake into cells after electroporation. We could describe the

uptake with the first-order model which corresponded to the dynamics of pore resealing in the cell

membrane. In several studies, speed of resealing was assessed and it was determined that it was in the range

of minutes in the in vitro as well as in the in vivo experiments, which was in agreement with our

experimental data (Kinosita and Tsong 1977; Saulis et al. 1991; Čemažar et al. 1998; Shirakashi et al. 2004;

Kandušer et al. 2006; Demiryurek et al. 2015).

In electroporation of excitable tissues, we can trigger action potential and thus muscle contraction and pain.

We determined experimental strength-duration curve for excitable and non-excitable cell lines and modeled

it with the phenomenological Lapicque curve and the Hodgkin-Huxley model. In future, it should be

established to what extent the in vitro determined strength-duration curve correlates to the in vivo muscle

contractions and pain sensation. When using strength-duration curves to predict action potential in excitable

cells, we have to take into account that for a while after the pulses cells are not able to trigger an action

potential. In future, a model describing the time between delivered electric pulses and action potential could

be determined. There are even studies which showed that high-voltage electric pulses could cause damage

to membrane proteins like voltage-gated channels (Chen et al. 2006b; Azan et al. 2017). The protein

conformational changes that open and close the channel gate take place on the order of a hundred times per

second, i.e., in the range of milliseconds (Gadsby 2009). It is interesting that nanosecond pulses are even

able to trigger an action potential (Pakhomov et al. 2017) as they are much shorter than the time needed for

conformational changes of voltage-gated channels. The models of excitability were determined

phenomenologically in the range of physiologically possible transmembrane voltages. However,

electroporation increases the transmembrane voltage more which is thus outside the range of the

physiological values. It should be established to what extent the current excitability models are valid in the

range of high transmembrane voltages (several hundreds of millivolts).

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Second, we have modeled the change in dielectric properties of tissues by modeling pore formation in cell

membranes and local transport region formation in the stratum corneum. In future, a similar model could

also be built for other tissues and validated with current-voltage measurements. For example, this could be

done for electroporation of liver metastases or muscles. A possible upgrade of the model is to model also

change in pore radii during pulses. Namely, with the asymptotic equation of pore formation, we assume that

all pores are of the same radii and only their density changes. We could use a method described in (Smith et

al. 2004) and model also change in pore radii. In this way, the change in of dielectric properties of skin

layers would not be determined based on the best agreement with the measurements, but it would be based

on mechanisms of electroporation. We achieved good agreement with current-voltage measurements when

long low-voltage pulses were applied. However, when short high-voltage pulses were applied the modeled

current deviated 50% from the measured one. In experiments, additional current component appeared which

could not be explained by the formation of the local transport regions. Even if we assumed that the whole

stratum corneum was one big local transport region or that the whole stratum corneum was substituted with

the electroporation buffer, the simulated current through the skin was lower than the measured one, and

simulated voltage drop on skin was too low as the conductivity of skin was too high. When measuring

voltage drop on the skin two thin copper wires were used, one on each side of the skin. Copper wires

created a preferential high-conductivity path for the electric current because they were much more

conductive than the electroporation buffer or the skin. Thus, electric current could locally increase and

contribute to the measured current, but it would not affect voltage drop on the skin. In future, also

electrochemistry happening during electroporation should be evaluated. During electroporation pH changes,

metal is released from the electrodes. We can see bubble formation, which could be a consequence of local

water evaporation or chemical reactions and formation of different gases. We could also include thermal

effects. Because of Joule heating the temperature increases and consequently, electric conductivity also

increases. In our model, heating was included indirectly, through the increase of radii of local transport

regions, which occurs due to lipid melting in the stratum corneum. In future, also melting of lipids could be

included in our model of local transport region formation (Becker and Kuznetsov 2007; Becker 2012;

Becker et al. 2014). Currently, we have taken the values of radii of local transport regions directly from

(Zorec et al. 2013a), we have only added a slight decrease in radii during the pause between pulses, which

described the experimental data better.

In the model, we have estimated the induced transmembrane voltage by treating the skin as a voltage

divider and taking into account the thickness of one layer and size of cells in that layer. However, the

induced transmembrane voltage is decreased due to close cell connections and electric field shielding. It is

possible that we overestimated the change in conductivity of lower layers and underestimated the

conductivity of local transport regions or their size and density. The change in the electric conductivity of

skin layers influences the electric field distribution and vice versa which could also introduce errors in our

modeling.

Third, we have modeled transport of small molecule in tumor during electroporation. Our model is valid for

small molecules in the range of cisplatin size (300 g/mol). It was shown that larger molecules pass cell

membrane with more difficulty than small molecules when electric pulses of the same parameters are

176


applied (Maček Lebar et al. 1998). The first reason could be lower diffusion coefficient of larger molecules

in the tumor interstitial fraction than of smaller molecules. Also, larger molecules enter cells predominantly

by electrophoresis while small molecules enter cells predominantly by diffusion (Pucihar et al. 2008).

Larger molecules, i.e., DNA, enter cells in several steps (Rosazza et al. 2016). Currently, in the model, we

assume that all the transport is due to diffusion, which is not necessarily always valid. Due to increased

interstitial pressure in the tumor (Baxter and Jain 1989; Boucher et al. 1990; Pušenjak and Miklavčič 2000),

convection mediated washout of drugs from the tumors is elevated. In the case of systemic drug

administration, this translates to the hindered entrance of drugs into the tumor. In the case of intratumoral

drug administration, which was employed in our in vivo experiments, this translates to increase in the

washout of the drug from the tumor. We found several velocities of the convective transport in the literature

(Ning et al. 1999). However, they did not correspond to our measurements of intracellular platinum mass

before pulse application and one hour after it. By inserting existing velocities from the literature in our

numerical model, all cisplatin was washed out in a few minutes. In future, the convective transport in the

mouse melanoma tumor needs to be evaluated. Our model was tested on only one tumor type. It is known

that different tumors have a different cell and matrix volume fractions (Mesojednik et al. 2007). Different

volume fractions influenced the diffusion coefficient of molecules in the tumor and the surface of the cell

membranes, available for the transport. In future, or model could be tested with different tumor types. We

could evaluate if transformation coefficient could be predicted from the tumor morphology. I believe that

such a transformation should exist – higher cell and matrix volume fractions in tumor should be reflected in

lower mobility of chemotherapeutic in the tumor and the cells and consequently, in lower transformation

coefficient. Another question is whether transformation coefficient depends on parameters of electric

pulses. Currently, we have tested our model at single pulse parameters in vivo. More experiments are

needed to test our model in wider parameter space and determine the functional dependency of the

transformation coefficient.

When delivering chemotherapeutic intratumorally, we avoid several steps in chemotherapeutic transport,

when it is delivered systemically. In systemic administration, the drug must first enter the bloodstream and

distribute through the vascular compartment, transport across the microvascular wall and then through the

interstitial fraction (Jain 1999; Jang et al. 2003). In a tumor, increased interstitial pressure decreases drug

transport into the tumor and facilitates drug washout from the tumor. In our model, we took into account

intratumoral delivery of cisplatin where we made a justified assumption of initial homogeneous distribution

of cisplatin in high concentration. If chemotherapeutic were delivered systemically, we would need to

additionally calculate initial conditions of cisplatin concentration in a tumor.

In the in vitro experiments, all cisplatin was accounted for. In the in vivo experiments, we injected 26 μg of

platinum by intratumoral injection of 80 μl in 35 mm3 tumors. The volume of injection was two-times the

volume of a tumor because we reused the data from (Uršič et al. 2018). Most of the injected platinum was

not measured. When a tumor was excised 2 min after cisplatin injection, we determined that 27 % of the

injected platinum was still present in a tumor and the serum. A part of cisplatin was lost due to lower tumor

volume than injection volume and due to washout of cisplatin from the tumor. When a tumor was excised

after 1 hour, we determined that in a tumor and serum there was 2% of the injected platinum (control

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samples - cisplatin injection, no electroporation) and 7 % in the treated (cisplatin injection and

electroporation) samples. Thus, in vivo, the majority of cisplatin was unaccounted for. Part of cisplatin was

lost when the tumor was mechanically disintegrated and during blood collection. Part of cisplatin could

accumulate in different organs and cause unwanted side effects (Sancho-Martínez et al. 2012). In future, it

should be established if injecting a lower volume of cisplatin could maintain high treatment efficacy while

decreasing unwanted side effects. In the literature, it was observed that changing the volume of injection did

not affect treatment efficacy (Wang 2006; Frandsen et al. 2017) which was also in agreement with our

results. Namely, although almost 70% of the injected cisplatin volume was lost already when injecting

cisplatin, the remaining cisplatin in the tumor was in high concentration, and consequently, the

concentration gradient was high.

Number of bleomycin molecules necessary to cause cell death was determined in (Tounekti et al. 1993;

Tounekti et al. 2001). A similar analysis should also be done for cisplatin as it would allow us to optimize

treatment parameters to achieve a high enough intracellular cisplatin concentration. The inductively coupled

plasma – mass spectrometry enables precise measurements of platinum mass in the cells. We could also

perform experiments evaluating cell death (e.g., by the clonogenic assay) at the same pulse parameters used

in the measurements of the platinum uptake. Then, we could link the survival and transport and determine

the necessary number of cisplatin molecules to cause cell death. Namely, the complete tumor eradication is

our final goal in electrochemotherapy, and our model of the cisplatin transport is a step forward in

achieving more precise prediction of cell death.

Electroporation results in an increase of membrane electric conductivity and permeability, which can be

understood as two separate phenomena (Miklavčič and Towhidi 2010; Leguèbe et al. 2014). The initial idea

was to connect membrane conductivity and permeability in one model. However, this idea was deemed

difficult to achieve. Already in the initial experiments, we determined that when exposing cells to electric

pulses in buffers which varied in their ionic composition but were of similar electric conductivity much

different time dynamics of a dye uptake was measured (Dermol et al. 2016). It was observed as well that

electroporation buffer conductivity did not affect cell sensitization (Gianulis et al. 2018). A similar electric

conductivity of a buffer would indicate similar induced transmembrane voltage and thus similar pore

density. In future, it would be interesting to determine what is the connection between cell membrane

conductivity and permeability or if permeability is largely influenced by biological processes happening in

a cell. The electric conductivity of cell membrane namely decreases relatively fast after pulse application,

while the transport lasts for several minutes or even hours if cells are left to reseal at lower temperatures

(Saulis et al. 1991). Another possible explanation of prolonged transport is the oxidative damage caused to

the cell membrane by electric pulses (Benov et al. 1994; Gabriel and Teissié 1995; Maccarrone et al. 1995;

Pakhomova et al. 2012). Oxidative damage disturbs the lipid bilayer and increases membrane permeability.

One of the main problems when modeling electroporation is the lack of measurements and experimental

data, e.g., electric conductivity of the local transport regions, the convective transport in the tumors, the

initial drug distribution prior to electroporation treatment, electric conductivity of separate skin layers

before and after electroporation, the change in local transport region radii as a function of pulse duration

and electric field. The models describing electroporation are namely very complex and have a large number

178


of parameters whose values are only estimated. When establishing and optimizing models of

electroporation, we can thus in many cases determine only local optimal values of parameters and not

necessarily global optimal values, as we determined in (Dermol-Černe and Miklavčič 2018).

In future, we aim to use the presented models in the treatment planning of electroporation-based treatments.

When optimizing electrode geometry and pulse parameters, currently we aim to reach a high-enough

experimentally determined critical electric field. We could first introduce the statistical models of cell death

and permeability. Later, we could aim to achieve a high-enough intracellular cisplatin concentration or even

number of molecules to cause cell death. We could decrease muscle contractions and pain which would

shorten and simplify the treatment. We could add measurement of the electric field in the tissue (Kranjc et

al. 2014; Kranjc et al. 2015) and change the pulse parameters in real-time while delivering electric pulses to

more precisely know the expected outcome.

In my opinion, natural sciences cannot exist without models which describe the processes, predict the

behavior of biological systems, offer insights into biological processes and their connections, and decrease

number of needed experiments. In this thesis, I have tried to unveil a small part of knowledge in the field of

electroporation. With the knowledge I have gained in the process of model development, even more

questions exist now than there did when I first started working on my Ph.D. and they should be answered in

future.

179

Original Scientific Contributions


Mathematical modeling of cell membrane permeability and cell death

We used mathematical models to describe and predict cell membrane permeability and depolarization and

cell death in vitro. First, we made an overview of the existing models. The existing models are mostly

phenomenological and do not include any electroporation mechanisms. Only a few models of cell

membrane permeability exist. On the contrary, there are many models of cell death because they are being

used for prediction of the efficacy of pulsed electric field assisted food pasteurization and sterilization. The

parameters of the models were optimized using experimental results. Cell membrane permeability and

depolarization were assessed by fluorescent dyes, and spectrofluorometric measurements and fluorescent

microscopy and cell death was assessed by metabolic activity test and clonogenic assay. The experiments

were performed on attached monolayers and cell suspensions. Our in vitro data was described well with

several different models. An easy transformation from cell level to tissue level was not possible – each

model had to be optimized separately for each tissue type, and thus their predictive power was limited.

The transition between single-cell models and the tissue models, where the realistic three-

dimensional skin model is presented with the model of electroporation of all components and

essential layers of the skin

Electroporation changes dielectric properties of tissues which can be assessed by different measuring

methods. We built a three-dimensional skin model by taking into account geometry of single cells. Each

cell was representative of the whole layer – corneocytes of the stratum corneum, keratinocytes of the

epidermis and lipid spheres of the papillary dermis. We modeled dielectric properties of each layer

analytically by the Hanai-Bruggeman formula or numerically, by a ‘unit cell’ to obtain a bulk skin model.

We added electroporation with long low-voltage or short high-voltage pulses to our model. We modeled

formation and growth of local transport regions in the stratum corneum. By asymptotic pore equation we

modeled pore formation on membranes of the cells of the lower layers. We accurately described current-

voltage measurements of skin electroporation. Our method can be used to determine the change in dielectric

properties due to electroporation of any tissue, as long as the geometric and the dielectric properties of cells

constituting the tissue are known.

Mathematical modeling of transport of small molecules (chemotherapeutic) across the cell

membrane using the dual porosity model

For efficient electrochemotherapy treatment, a sufficient uptake of chemotherapeutic in the tumor cells has

to be achieved to cause cell death. We accurately described the uptake of cisplatin by the tumor cells in the

mouse melanoma tumors. We performed experiments in vitro and measured the mass of the intracellular

platinum by the inductively coupled plasma – mass spectrometry. With the dual porosity model, we

calculated the permeability coefficient as a function of pulse number and electric field. We have built a 3D

tumor model where we described the transport of cisplatin between several compartments - tumor cells,

interstitial tumor fraction, and peritumoral environment. We determined that in vitro permeability

coefficient could be used to model uptake of cisplatin in vivo. However, in vivo, the transport was slower

due to close cell connections, smaller available membrane area for transport and presence of cell matrix.

180


These differences between in vivo and in vitro were taken into account by a transformation coefficient,

which is most likely a function of tumor morphology. Our model can be in future used in treatment

planning of electrochemotherapy.

181

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